INTEGRATION OF GEOMORPHIC EXPERIMENT DATA IN … · 2015-04-09 · Dynamic process distributing...

INTEGRATION OF GEOMORPHIC EXPERIMENT DATA IN

SURFACE-BASED MODELING: FROM

CHARACTERIZATION TO SIMULATION

A DISSERTATION

SUBMITTED TO THE INTERDISCIPLINARY PROGRAM OF

EARTH, ENERGY, AND ENVIRONMENTAL SCIENCES

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Siyao Xu

March 2014

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/rj323qf2670

© 2014 by Siyao Xu. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

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http://purl.stanford.edu/rj323qf2670

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Tapan Mukerji, Primary Adviser


Jef Caers


George Hilley

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

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v

ABSTRACT

Dynamic process distributing geobodies, defined as geological dynamic processes

in this dissertation, are the determinants of spatial heterogeneity of petrofacies, which

further affect reservoir performances. But a dynamic process is normally not considered

by conventional static reservoir modeling techniques, e.g. two-point statistics, multiple

point statistics, and object-based methods. The surface-based method is the only static

reservoir modeling technique that explicitly considers a geological dynamic process.

Moreover, the surface-based method is an open and flexible framework that is capable

of integrating various techniques to generate realistic model realizations. However,

current implementations of geological dynamic processes in surface-based models rely

on imprecise conceptual rules, which normally involve a large number of empirical

coefficients and subjective decisions to quantify qualitative concepts. This does not

appear to be an ideal treatment, since workloads in reservoir uncertainty estimation are

increased along with empirical coefficients and subjective decisions. The demand of an

improved implementation of geological dynamic processes that is more proper for the

Monte Carlo uncertainty estimation led to the research in this dissertation. Contributions

of this dissertation include a treatment to extract erosion rules from the records of an

experiment, a solution to identify a geomorphic experiment to a real depositional system,

and a new strategy to implement a geological dynamic process in surface-based models

using geomorphic experimental data.

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The shale drape coverage in deepwater environments is a key parameter to the

sand body connectivity in a reservoir. Proper modeling of shale drape coverage requires

understanding of erosion geometries, which must be extracted from changes of

intermediate topography of the depositional process. However, intermediate topography

is normally not maintained in the stratigraphy of real depositional systems, thus

understanding on the erosion geometry is always limited. To study this issue, we

developed a workflow based on a geomorphic experiment, where the intermediate status

was available in the records. An experiment of a delta basin with recorded intermediate

topography was used to demonstrate this workflow. Sequential correlated patterns of

erosion and deposition were extracted from intermediate topographies. A surface-based

model of lobes and distributary channels was built based on input statistics of

experimental erosion-deposition geometries. Since the geomorphic experiment was at a

different scale than any real scale depositional system, we proposed to measure the

cumulative distribution functions of dimensionless ratios to characterize correlations

between erosion depth and deposition thickness in the experiments.

Static reservoir models in the oil industry are always required to be specified to

the real scale system of a reservoir. Thus, the selected experiment to provide information

for a surface-based model is required to be as similar to the system of the reservoir as

possible. However, no hydrodynamic or stratigraphic method has been developed to

estimate similarities between an experiment and a specific real depositional system. A

solution is provided to identify one experiment that is most similar to a given real system

from a set of optional experiments. The solution estimates a similarity between lobe

stacking patterns of two systems, which are characterized by the cumulative distribution

functions of pairwise lobate proximity measurements. The similarity is estimated with

a bootstrap two-sample hypothesis test on the two cumulative distribution functions.

Since lobate bodies in experiments can be identified hierarchically from small scales to

large scales depending on decisions of the interpreter, the solution also includes an

automatic method to quantify lobe hierarchies and to choose lobate stacking patterns at

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various scales of interpretation. This solution is applied to estimate the similarity

between two delta fan experiments and two published real systems.

Based on statistical similarity analysis, an experimental lobe pattern was

identified as the pattern with the highest similarity to a real system. A natural

assumption is that the forward process forming identified experimental lobate pattern is

also similar to the process forming the given real lobate pattern. In this sense, an

experimental lobe pattern is a prior scenario of the spatial-temporal lobe distributing

process and is input to the surface-based model. A new surface-based model for a lobate

environment was developed, in which the lobe migration mechanism was implemented

based on the correlated random walk. The input lobe pattern was treated as a sequence,

characterized by statistics of migration distance and migration orientation shifting angle.

Both control the random walk behavior of lobe migration. Only three cumulative

distribution functions from the input pattern and two empirical coefficients were

introduced by the random-walk-based lobe migration mechanism. Compared to tens of

empirical coefficients, the new mechanism was more appropriate for uncertainty

estimations using a Monte Carlo method. Demonstrations also show that realizations of

the new model were hierarchically similar to the input lobe sequence analogous to the

similarity between realizations of multiple point statistics and the training image.

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ACKNOWLEDGEMENTS

Turning back and recalling the past four years, I would like to express the most

sincere gratitude to my advisor Professor Tapan Mukerji, who has been supporting and

guiding me through my time in Stanford. When I came here as a general geoscientist

and software engineer, Tapan tolerated my naïve questions with his great patience and

led me to the correct path. Tapan is knowledgable. Every time an obstacle appeared in

my research, a conversation with Tapan was the best way to organize my chaotic mind,

refine the problem, and locate a good solution. He has also been a good friend when I

have had difficulties in life.

I would also like to express my gratitude to Professor Jef Caers, whose active

mind is one of the richest sources of ideas. Jef’s talk and comments in every seminar

introduced his insightful opinions on research, guiding me toward meaningful and

promising directions. My work would not be in the current form without Jef.

I would like to appreciate all those I have been talking to through the duration of

my Ph.D., including Tim McHargue, George Hilley, and Gary Mavko. Tim’s precious

experiences from the very frontier of Surface-based modeling and critical comments

from George and Gary were the base pillars of successful research.

My appreciation also goes to Professor Chris Paola and his students: Antoinette

Abeyta, Sarah Baumgardner, and Dan Cazanacli. Without Chris’ data and valuable

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responses, our ideas at SCRF would not be implemented. It was my honor to witness

and contribute to the beginning of a successful collaboration between SCRF and SAFL.

I also benefitted quite a lot from my three internships at Shell. Names I must

mention are Omer Alpak, J-C Noirot, Matt Wolinsky, Long Jin, Tianhong Chen, and

Paul Gelderblom. Any work and conversation with you enhanced my understanding

about the industry, which was the required background for my research.

I am thankful to my colleagues and friends in SCRF: Andre Jung, Orhun Aydin,

Lewis Li (please forgive me for not listing all of them here), from whom I have always

received both help and fun. We have built a perfect community for work and life. It is

my honor to be a member of SCRF.

I also appreciate my best friend Yang Liu, who was my unofficial mentor. I have

learned much from him in research and in stories of this industry, and I will always

remember other fun memories we have had, for example, those tickets.

I would like to say thanks to Professor Milton E. Harvey, who granted me selfless

help through my two years in Ohio. I will never forget our discussions, covering topics

from research to baseball and from politics to food.

I have always been thankful to live and study within the community of

Department of Energy Resource Engineering, where you can always find helpful friends.

I dedicate this dissertation to Dr. Ning, my sweetie. I will bear in mind that

Christmas, when the primary idea of this work and our wedding came true in Redwood

City, the most romantic place in the Bay Area.

This dissertation is also dedicated to my mum, dad, and grandma.

Siyao Xu

Stanford University

March, 2014

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TABLE OF CONTENTS

ABSTRACT ............................................................................................................................................. V

ACKNOWLEDGEMENTS ............................................................................................................... VIII

TABLE OF CONTENTS ........................................................................................................................ X

LIST OF TABLES ............................................................................................................................... XII

LIST OF FIGURES ........................................................................................................................... XIII

CHAPTER 1 INTRODUCTION ......................................................................................................... 1

1.1 REVIEW OF STATIC RESERVOIR MODELING ALGORITHMS ........................................................... 3 1.2 CHOOSING MODELING ALGORITHMS ............................................................................................ 5 1.3 SURFACE-BASED MODELING AND CHALLENGES .......................................................................... 8 1.4 THE PROPOSED APPROACH ......................................................................................................... 12 1.5 THESIS OUTLINE ......................................................................................................................... 14

CHAPTER 2 EROSION RULES IN SURFACE-BASED MODELING: GEOMORPHIC

EXPERIMENTS AS THE KNOWLEDGE DATABASE .................................................................. 15

2.1 INTRODUCTION ........................................................................................................................... 15 2.1.1 Model Setting: Deepwater Turbidite System ........................................................................... 15 2.1.2 Erosion in Static Reservoir Modeling .......................................................................................... 17

2.2 METHODOLOGY .......................................................................................................................... 21 2.2.1 Surface-Based Modeling .................................................................................................................... 21 2.2.2 Building a Distributary Channel-Lobe Systems ...................................................................... 24 2.2.3 Geomorphic Experiments ................................................................................................................. 35

2.3 SIMULATION RESULTS ................................................................................................................ 43 2.4 CHAPTER SUMMARY ................................................................................................................... 47

CHAPTER 3 THE STATISTICAL EXTERNAL SIMILARITY BETWEEN LOBATE

ENVIRONMENTS: LINKING EXPERIMENTS TO REAL-SCALE SYSTEMS ......................... 48

3.1 INTRODUCTION ........................................................................................................................... 48 3.2 METHODOLOGY .......................................................................................................................... 50

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3.2.1 Experimental Data and Preprocessing ........................................................................................ 51 3.2.2 Stage One – Automatic Lobe Hierarchy Quantification ........................................................ 54 3.2.3 Stage Two – Testing Pattern External Similarity .................................................................... 67

3.3 APPLICATION .............................................................................................................................. 75 3.4 CHAPTER SUMMARY .................................................................................................................. 83

CHAPTER 4 HIERARCHICAL SIMILARITY OF LOBES: SIMULATION AND

COMPARISON ...................................................................................................................................... 85

4.1 INTRODUCTION ........................................................................................................................... 85 4.2 A LOBE MIGRATION MODEL WITH INPUT LOBE SEQUENCE ....................................................... 88

4.2.1 Root Mean Square Distances: The Scaling Factor for Stacking Patterns ...................... 89 4.2.2 The Bifactor Modeling Scheme ....................................................................................................... 90 4.2.3 The Input Stacking Pattern Factor ................................................................................................ 92 4.2.4 The Modeling Algorithms .................................................................................................................. 97 4.2.5 Example ..................................................................................................................................................... 99

4.3 HIERARCHICAL SIMILARITY ..................................................................................................... 105 4.3.1 Quantitative Measure of Hierarchical Similarity .................................................................. 106 4.3.2 Implicit Hierarchical Control in the CRW-based Mechanism ......................................... 112 4.3.3 Application of Hierarchical Similarity Control ...................................................................... 117

4.4 CHAPTER SUMMARY ................................................................................................................ 119

CHAPTER 5 CONCLUSIONS AND FUTURE WORK .............................................................. 121

5.1 CONCLUSIONS .......................................................................................................................... 121 5.2 RECOMMENDATIONS FOR FUTURE WORK ................................................................................ 124

BIBLIOGRAPHY ................................................................................................................................ 129

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LIST OF TABLES

Table 2.1: Three categories of patterns are interpreted from the sediment event plot 1) channels,

where the width of erosion is equal or greater than that of deposition; 2) lobe with

erosion, where the width of deposition is greater than that of erosion; 3) lobe without

erosion. ........................................................................................................................ 42

Table 2.2: Dimensionless ratios are used to characterize the relationship between erosion-deposition

geometries identified from the sediment event plot. Two probability functions

respectively for channel and for lobe with erosion are interpreted. The probability of

erosion occurs with lobes is 0.29. ................................................................................ 43

Table 2.3: Primary simulation parameters. ....................................................................................... 44

Table 3.1: P-values of hypothesis testing between samples. Sample 1 and 2 are detected to be similar,

which Sample 3 is different from 1 and 2. ................................................................... 74

Table 4.1: Values of primary parameters in the simulation. ........................................................... 101

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LIST OF FIGURES

Figure 1.1: Various static reservoir modeling techniques. Along with the improvement of geological

realism from two-point statistics to process-based method, the difficulty of

conditioning models increases. After Bertoncello 2011. .............................................. 7

Figure 1.2: A combined modeling algorithm. Different algorithms are used at proper stages to

maximize their advantages. After Michael et al. 2010. ................................................. 8

Figure 2.1: The demonstration of a deepwater turbidite system. The model setting of this work is in

the lower fan section of this system (red square). ....................................................... 16

Figure 2.2: A demonstration of the hierarchical structure of lobate systems. The nomenclature

follows Prélat et al. 2010. ............................................................................................ 17

Figure 2.3: The repetitive appearance of fine grained interlobe layers. After Prélat, Hodgson, and

Flint 2009. ................................................................................................................... 19

Figure 2.4: The eroded shale drapes from an outcrop. After Alpak, Barton, and Castineira 2011. . 20

Figure 2.5: The proposed strategy of using geomorphic experiment data in surface-based models.

Experimental data are used to extract geometry and depositional rules in Step 4. Steps

1-3 refers to Bertoncello 2011. .................................................................................... 20

Figure 2.6: The general forward workflow for surface-based models. After Caers, 2012. .............. 23

Figure 2.7: The template boundary of a distributary channel-lobe geobody interpreted from overhead

photos. The channel is interpreted to be constant width and the lobe is interpreted to be

an oval. Four centerline control points with geometric or geological interpretation are

defined to determine the morphology of a specific geobody. ..................................... 25

Figure 2.8: Parameters controlling template boundary for a lobe. ................................................... 26

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Figure 2.9: The channel trend map generated by normalized perpendicular distance from centerline

to the boundary. ........................................................................................................... 27

Figure 2.10: The lobe trend map generated by normalized distance from MPP to the lobe

boundary. ..................................................................................................................... 27

Figure 2.11: The concept of positive-negative surfaces for erosion in surface-based models: a) a new

lobe is generated and placed onto the previous topographic surface; b) due to erosion,

a portion of the new lobe is below the previous topography; c) positive surface: the

portion of the new lobe above the previous topography; negative surface: the portion

of the new lobe below the previous topography. ......................................................... 28

Figure 2.12: Two pairs of positive-negative surfaces. The dimensionless ratio of maximum erosion

depth versus maximum deposition thickness is 3

7. ....................................................... 29

Figure 2.13: Parameters to calculate positive-negative surfaces for a) channel and b) lobe using trend

map and trend-to-thickness functions in Eq. (2.5) – (2.8). .......................................... 30

Figure 2.14: Locations of the centerline control points of a geobody are determined following the

order 1) channel source 2) MPP 3) channel turning 4) lobe distal point. ................... 33

Figure 2.15: The channel source is assumed to be uniformly distributed within a segment around the

midpoint of an edge of the modeling grid. .................................................................. 33

Figure 2.16: The depositional rules for the second control point: lobe Most Proximal Point (MPP).

The MPP is drawn from a combined conceptual two-dimensional probability

distribution on the modeling grid. The distribution is the weighted mean of three

components: a) the next MPP (black dot) tends to be close to the source (star), so for

each point in the grid, the probability is proportional to the reciprocal of its distance to

a predefined basin source; b) the next MPP (black dot) tends to be close to a previous

MPP (blue dot); c) the next MPP (black dot) tends to appear in the region with fewer

deposits; d) the final two-dimensional probability map is a weighted mean of the three

components. ................................................................................................................. 34

Figure 2.17: The depositional rule for channel turning. Once the channel source A and lobe most

proximal point C are determined, a preset window is placed around the midpoint of

Segment 𝐴𝐶̅̅ ̅̅ . The conceptual probability within the window is determined by the

normalized reciprocal distance from the previous channel turning G to every cell within

the window. The new channel turning B is sampled within the window. ................... 34

Figure 2.18: The experiment setting for DB-03. The experiment was performed in a tank of five

meters by five meters, in which the infeed point was set at a corner of the tank. A prerun

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was performed to generate the initial topography, which was a symmetric delta with

the radius of 2.5 meters apart from the infeed point. .................................................. 37

Figure 2.19: The intermediate topography of the experiment was measured along three lines at 1.5

meters, 1.75 meters, and 2 meters apart from the infeed point, respectively. ............. 37

Figure 2.20: A fraction of the final stratigraphy measured on one topography line, in which the

intermediate conditions are partially removed by following events. .......................... 38

Figure 2.21: The erosion and deposition are identified by computing differences between two

successive records. Erosion is defined by reduced elevation between two successive

elevation records, while deposition is defined by increased elevation. ....................... 38

Figure 2.22: Examples of generating a sediment event plot from intermediate topography records: a)

to identify the relative elevation change regions between two successive intermediate

topography records (the upper plot). The relative change geometry is plotted as its

temporal order (the lower plot); b) to identify the series of geometries of the elevation

change and plot the all geometries along the vertical axis as the order of

occurrence. .................................................................................................................. 41

Figure 2.23: The sediment event plot of 100 records on the topography line at 1.75 meters apart from

the infeed point. ........................................................................................................... 42

Figure 2.24: The intermediate steps of the simulation for the first four lobes. Two cross sections are

used to monitor stratigraphic changes at every step. Erosion does not occur in Lobe 1

while Lobe 2 – Lobe 4 all cause erosion. .................................................................... 45

Figure 2.25: A realization of 30 lobes. Two cross sections along and perpendicular to the basin flow

orientation are also shown. .......................................................................................... 46

Figure 2.26: A fractions of the experimental and simulated sequential pattern plots. Geometry of

erosion and deposition are different, since no three-dimensional data are available to

calibrate the trend-to-thickness function. The occurrence of paired negative-positive

surfaces for the channel and lobes (with and without erosion) are observed in the

simulated result, as expected. ...................................................................................... 46

Figure 2.27: The QQ plots prove the reproduction of cumulative distribution functions of the

dimensionless ratios in the model realization. ............................................................. 47

Figure 3.1: Corrected overhead photos of an experiment taken at regular intervals, in which only one

active channel was recorded within the successive photos. Courtesy by Prof. Chris

Paola, San Anthony Falls National Lab. ..................................................................... 53

Figure 3.2: The lobe connected to the latest channel was interpreted on every photo, and a series of

lobes are obtained. ....................................................................................................... 53

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Figure 3.3: Parameterization of lobe stacking patterns: a) the source is defined as the MPP on the

lobe boundary to the basin source; b) lobe orientation is defined by the unit vector

pointing from the source point to the geometric centroid of a lobe; c) polygonal

proximity is measured by the Haussdorff distance between two lobes; d) shape

proximity is measured through procrustes analysis. .................................................... 56

Figure 3.4: Synthetic demonstration of the temporal constraint: a) based on rules in the interpretation,

only agglomerations on temporarily successive lobes are legitimate in clustering; b) the

sequential cumulated distance from Lobe 𝑡1 to Lobe 𝑡3 is the distance following the

sequential route from t1 to t2 to t3, promising that d13 is always greater than d12 and

d23. ................................................................................................................................ 62

Figure 3.5: The hierarchical clustering result is visualized by a dendrogram. In the synthetic

demonstration, distance 𝑑12 < 𝑑23 < 𝑑13 , the dendrogram visualizes a lower scale

group (𝑡1, 𝑡2) and a higher scale group ((𝑡1, 𝑡2), 𝑡3). ................................................... 62

Figure 3.6: The dendrogram of a complex dataset. Visualization of large datasets with dendrograms

is inconvenient. ............................................................................................................ 63

Figure 3.7: The reachability plot as a dendrogram alternative to visualize a large dataset. The height

of the bars represent the distance from one observation to all the left neighbors in the

plot. The first bar is set to be the maximum height within this clustering. ................. 63

Figure 3.8: The threshold intersects with bars greater than the chosen scale of interpretation within a

reachability plot. Lobes between two intersections form a lobe at the chosen scale. .. 64

Figure 3.9: A sample lobe sequence of 36 lobes. Color represents lobe location within the sequence,

ranging from 1 to 36. ................................................................................................... 65

Figure 3.10: a) The dendrogram of the sample lobes, the 36 sample lobes are organized by the order

of being clustered in the clustering algorithm, and listed along the horizontal axis; b)

The reachability plot of the sample lobe; lobes are listed with the same order as in the

dendrogram. ................................................................................................................. 66

Figure 3.11: Clusters on the exemplary sample of lobes with a selected threshold. Lobes in each

group form a lobe at the scale of interpretation. .......................................................... 67

Figure 3.12: a) Exemplary set of 60 lobes, which is a super set of the sample lobe set in Fig. 3.9. b)

The point process of MPP of lobes in Fig. 3.12 a. ....................................................... 69

Figure 3.13: G functions (normalized using procrustes analysis) of the spatial pattern in Fig.

3.12. ............................................................................................................................. 70

Figure 3.14: a) Lobe Sample 1 of 13 lobes; b) Sample 2 of 16 lobes; c) Sample 3 of 13 lobes. Sample

1 and Sample 2 are from the same depositional system. ............................................. 73

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Figure 3.15: 𝐺 functions for all three samples, the samples are normalized using procrustes analysis

so length measurements from different systems are comparable. ............................... 74

Figure 3.16: The workflow of scale-dependent similarity analysis. The result is a p-value versus a

scale plot for each parameter. ...................................................................................... 77

Figure 3.17: Lobe patterns used in the application: a) stacking pattern of Exp. A; b) stacking pattern

of Exp. B; c) stacking pattern of Kutai basin; d) stacking pattern of Amazon fan. Arrows

mark out the overall flow orientations for experiments and the interpreted overall flow

orientation for the real-scale cases.. ............................................................................ 77

Figure 3.18: Hierarchies of experiments: a) hierarchy of Exp. A; b) hierarchy of Exp. B. .............. 78

Figure 3.19: P-values versus scales of interpretation plots with respect to the Kutai basin. The

distance measurements are rescaled with RMSD of the Kutai basin. Peaks of p-values

are marked out by the black dots and a narrow range for scales of interpretation with

the highest p-values for all four parameters (shaded regions) is identified. a) MPP; b)

Orientation; c) Polygonal distance; d) Shape. ............................................................. 80

Figure 3.20: P-values versus scales of interpretation plots with respect to the Amazon fan. The

distance measurements are rescaled with RMSD of the Amazon fan. Peaks of p-values

are marked out by the black dots, but the peaks vary in a wide range of scales. a) MPP;

b) Orientation; c) Polygonal distance; d) Shape. ......................................................... 81

Figure 3.21: Stacking patterns from Exp. B at peaks of p-values with respect to the Kutai basin: a)

patterns at the scale of interpretation with the highest p-value for the MPP (38 lobes) at

the scale of 13.8km; b) patterns at the scale of interpretation with the highest p-value

for 𝜃 (35 lobes) at the scale of 15.1km for; c) patterns at the scale of interpretation with

the highest p-value for polygonal distance (30 lobes) at the scale of 15.8km; d) patterns

at the scale of interpretation with the highest p-value for shape (28 lobes) at the scale

of 16.8km. ................................................................................................................... 82

Figure 3.22: Stacking patterns from Exp. B at peaks of p-values with respect to the Amazon basin:

a) patterns at the scale of interpretation with the highest p-value for the MPP (22 lobes)

at the scale of 62km; b) patterns at the scale of interpretation with the highest p-value

for 𝜃 (26 lobes) at the scale of 47km for; c) patterns at the scale of interpretation with

the highest p-value for polygonal distance (30 lobes) at the scale of 42km; d) patterns

at the scale of interpretation with the highest p-value for shape (65 lobes) at the scale

of 14km. ...................................................................................................................... 82

Figure 4.1: a) The plane view of an experimental stacking pattern with the high p-values in Chapter

3, including 38 lobes. b) The sequence view of the experimental stacking pattern, which

is explicitly input to the algorithm in this chapter. ...................................................... 89

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Figure 4.2: Root Mean Square Distance (RMSD) as the scaling factor for stacking patterns. RMSD

is determined by the averaged squared distance from the geometric centroid to the

interpreted basin boundary. It is a measure of the scale of a stacking pattern and is used

as the scaling vector to normalize stacking patterns. All length measurements and

coordinates from the experiment are normalized by the RMSD of the experimental

pattern, and the simulation is performed in the dimensionless space. The RMSD of the

field basin can be used to scale the realizations to the basin scale. Refer to Eq. (4.1) –

(4.3). ............................................................................................................................. 90

Figure 4.3: MPP migration and lobe relative (to the basin flow orientation) shifting angle are

modeled in this chapter. Scale versus stacking pattern plots of the area distance and the

shape are noisy, thus the use of them as inputs requires further studies. a) ∆𝑟, the

stepwise migrating distance of MPP; b) ∆𝜃𝑀𝑃𝑃 , the stepwise MPP migrating

orientation, measured from 0 to 2π; c) ∆𝜃𝐿, the stepwise lobe relative (to the basin flow

orientation) orientation, measured from 0 to π............................................................ 94

Figure 4.4: The idea of Correlated Random Walk (CRW). CRW simulates spatial point sequences.

The point at time 𝑡 + 1 is determined by deviations from the point at 𝑡 by a moving

distance ∆𝑟 and an angle ∆𝜃𝑀𝑃𝑃 deviating from the previous MPP moving orientation.

∆𝑟 and ∆𝜃𝑀𝑃𝑃 are modeled as stochastic variables with cumulative distribution

functions (CDFs). The choice of CDFs depends on the problem. CRW is applied to

model MPP migration in this work and controlled by empirical CDFs from the input

lobe sequence. .............................................................................................................. 95

Figure 4.5: The randomness in CRW. The migration distance ∆𝑟 is usually modeled by a weibull

distribution and ∆𝜃𝑀𝑃𝑃is modeled by a wrapped cauchy distribution. Both distributions

provide parameters to control the means and shapes. Distributions with higher

probability to large ∆𝑟 and ∆𝜃𝑀𝑃𝑃 lead to more randomly distributed point sequences,

while higher probability for small ∆𝑟 and ∆𝜃𝑀𝑃𝑃 lead to sequences consistently

migrating in the same direction. .................................................................................. 97

Figure 4.6: The implementation of the combined lobe migration mechanism: a) the bifactor model

combining MPP migration information from the input lobe sequence and the conceptual

compensational stacking rule in response to the intermediate topography in the

simulation, refers to Algorithm 4.1; b) determining the lobe orientation, refers to

Algorithm 4.2. .............................................................................................................. 99

Figure 4.7: Model grid setting for the test simulation. The basin source sample within a small section

around the middle point of an edge (refer to Table 4.1) and the basin flow orientation

xix

were set to be perpendicular to the edge of the basin source toward the modeling

grid. ........................................................................................................................... 101

Figure 4.8: ECDFs from the inputs lobe migration sequence. ........................................................ 102

Figure 4.9: Samples of intermediate topographies of a realization with 100 lobes. Several lobe

complexes following the compensational stacking rule are observed in the

simulation. ................................................................................................................. 103

Figure 4.10: One realization of 100 lobes and the proximal, medial, and distal depositional sequence

plot. Lateral shifting and spatial-temporal clustering are observed. ......................... 104

Figure 4.11: The reproduction of input ECDFs of ∆𝑟, ∆𝜃𝑀𝑃𝑃, and ∆𝜃𝐿𝑜𝑏𝑒. Relative MPP migrating

orientation ∆𝜃𝑀𝑃𝑃 is not reproduced due to the combination with in-situ compensation

stacking rules. ............................................................................................................ 105

Figure 4.12: MPP migration sequence of the input stacking pattern: a) the plain view; b) the

perspective view. ....................................................................................................... 109

Figure 4.13: MPP migration sequence of the realization: a) the plain view; b) the perspective

view. .......................................................................................................................... 109

Figure 4.14: The sequential hierarchies of input and realization. The sequential hierarchy is

quantified by the dendrogram with the sequential cumulated distance. The hierarchical

similarity is defined by the similarity between two dendrograms. a) The input stacking

pattern; b) The realization. ........................................................................................ 110

Figure 4.15 A dendrogram is a weighted binary tree with three components: topology, label, and

weights. a) topology: the bifurcation structure between leaves of the tree; b) label: links

each leaf to a branch; c) weights: the height at which two branches merge, implying the

distance between two clusters represented by the two branches. .............................. 111

Figure 4.16: Cophenetic distance: distance at which two leaves merge. In the example, the cophenetic

distance from Leaf 1 to Leaf 3 is 2.5. ........................................................................ 111

Figure 4.17: ECDFs of the cophenetic distance from the input lobe sequence and from the simulated

realization in Fig. 4.14. .............................................................................................. 112

Figure 4.18: The linkage function, a strategy of calculating the intercluster distance in the

agglomerative hierarchical clustering, determines the ECDF of the cophenetic distance.

An imprecise relationship exists between the ECDF of the stepwise migration distance

and the ECDF of the cophenetic distance from the dendrogram. ............................. 115

Figure 4.19: The denrograms generated with different linkage functions using the input lobe sequence

in Fig. 4.12 and the realization in Fig. 4.13. The visual comparison can easily identify

the difference between the dendrograms with different linkage function on the same set

of data. However, no significant difference is observed between dendrograms of the

xx

input and realizations with the same linkage function. The L1-norm differences

between the ECDFs of the cophenetic distances of the input and realization

dendrograms are calculated for each linkage function: 3758.4 for complete linkage,

1566.6 for average linkage, and 899.3 for single linkage, which is consistent with the

theoretic analysis. ...................................................................................................... 116

Figure 4.20: First two component multidimensional scaling plot of hierarchies of realizations,

demonstrating the effect of an input lobe sequence on constraining the MPP hierarchies

of model realizations, which is difficult with the model built in Chapter 2. ............. 118

1

Chapter 1

INTRODUCTION

Static reservoir modeling is an information integration procedure that aims to

produce geologically reasonable model realizations for dynamic reservoir simulations

and, as a result, can be used to estimate reservoir performance uncertainty. The

information includes soft and hard data plus qualitative geological knowledge. Hard

data are logs and cores measured around wells and provide precise information. Soft

data are seismic surveys performed at large regions with coarser spatial and temporal

resolution, such that the provided information is considered averaged and smoothed

over a unit volume of rocks. Qualitative geological knowledge is provided by field

surveys on outcrops analogous to the depositional environment of a reservoir,

containing direct field measurements and interpretations based on measurements. The

direct measurements include structural and stratigraphic architectures and petrophysical

properties. Interpretations provide conceptual models of the spatial distribution of

petrofacies in the studied depositional environments or even the ‘dynamic’ depositional

process driving the spatial distribution of petrofacies. A static model used for

hydrocarbon exploration and development is required to reasonably integrate all

2

available information such that any features affecting the subsurface flow movement

and hydrocarbon production are realistically reproduced.

However, all of the three types of information are limited. The hard data are at

very fine scales compared to the reservoir, such that only a tiny fraction of the

subsurface is revealed. The scale of the seismic survey is too coarse to describe

stratigraphies at the scale of reservoirs. Interpretation may vary from one interpreter to

another, creating uncertainty in estimations. In the context of uncertainty estimation and

history matching, static modeling algorithms are supposed to be simple and

computationally efficient. A simple and fast static model reduces the workload of

uncertainty estimations and history matching. In short, the core of static reservoir

modeling is to build models as geologically realistic as possible while the model

simplicity and computational efficiency are considered. Various static modeling

algorithms have been devised in earlier publications, some of which populate

petrophysical properties in modeling domains that account for the statistical spatial

correlation from the data. Others attempt to use the real underlying physics of

depositional processes. However, the choice of modeling techniques is problem-

dependent. The focus of this thesis is the surface-based method, which attempts to

explicitly emulate realistic spatial distribution in interpreted geometric bodies of

petrofacies and concurrently avoid inefficiencies in solving hydrodynamic equations. In

Section 1.1, we will review major techniques of static reservoir modeling. Then, in

Section 1.2, the general surface-based framework of using different algorithms to

generate satisfying static models is described. Finally, in Section 1.3, challenges in

surface-based method are discussed, leading to the following chapters of this

dissertation.

3

1.1 Review of Static Reservoir Modeling Algorithms

Various techniques have been developed for static reservoir modeling in recent

decades. A general trend in the development of static modeling algorithms is the

increment of explicit geology accounting in the algorithms and the accompanied

increment in algorithm complexity.

Reservoir engineers and geophysicists developed algorithms in the two-point

geostatistics toolbox, focusing on the spatial distribution of petrophysical properties

important for dynamic reservoir simulation. Spatial correlation of petrophysical

properties is characterized by the variogram (Goovaerts 1997), which is, in turn, used

to control the simulation. Variogram is the quantitative form of geology in two-point

geostatistics and the unit to be simulated is a property value in a cell, so it is impossible

to model complex geometries. Two-point geostatistics is simple and efficient, thus it

has been widely utilized in the industry. Multiple-point geostatistics (Strebelle 2002;

Zhang, Switzer, and Journel 2006; Arpat and Caers 2007) is an improved technique

developed by reservoir engineers as well, aiming at simulating more complex

heterogeneities. Multiple-point statistics also works on property values in cells. As an

alternative to variograms and correlations between pairwise values, multiple-point

statistics uses training images to characterize the prior geological information and the

cell value correlation is estimated using several surrounding points. A training image is

originally a sketch of conceptual spatial distributions of petrofacies. Realizations of

multiple-point statistics are visually plausible to the training image, while conditioning

remains easy due to the cell-based nature of the algorithm. Advancement in multiple-

point statistics can generate realizations with nonstationary training images (Honarkhah

and Caers 2012).

The idea of the object-based method (Shmaryan and Deutsch 1999; Deutsch and

Wang 1996; Deutsch 2002) is closer to the geologist’s perspective than that of the

engineer. The method works at the scale of interpreted geometric objects of petrofacies,

or geobodies. Geobodies are placed stochastically into the modeling grid with spatial

4

constraints, such as adjacency of different geobodies or trends of the spatial distribution

of geobodies, until a predefined criterion is satisfied. Since geobodies constitute explicit

geometric information, complex patterns are produced by the object-based method.

However, conditioning of object-based models to hard data is limited to trial-and-error,

which is extremely computationally expensive.

Process-based models attempt to solve the problem from the perspective of

physical laws. Hydrodynamic equations are used to describe the underlying physics of

depositional processes. The process-based model exceeds other methods in terms of

realism. However, since the equations are solved numerically over a fine grid in order

to provide geological details, process-based models are the most time consuming and

not directly applicable to reservoir modeling. Moreover, there is no information to

estimate physical parameters of a paleoflow, such as discharged, sediment

concentrations that occurred millions of years ago etc., thus realizations of process-

based models are normally used as three dimensional general references to improve

geological understanding that is not available in real depositional systems.

Similar to object-based modeling, surface-based modeling (Pyrcz, Catuneanu,

and Deutsch 2005; McHargue et al. 2011; Pyrcz and Strebelle 2006; Pyrcz et al. 2012;

Michael et al. 2010; Miller et al. 2008) also explicitly models the geometric information

of petrofacies to generate complex realizations. Nevertheless, more complex

depositional rules are utilized to mimic deposition-erosion processes in a forward

scheme, such that the final stratigraphy is plausible to the results of process-based

methods. Since no equations are solved, complex stratigraphy can be generated at much

lower computational costs with surface-based methods. The surface-based model is a

forward modeling method, thus an inversion-based conditioning algorithm (Bertoncello

2011; Bertoncello et al. 2013), which is time-consuming, is required.

5

1.2 Choosing Modeling Algorithms

As indicated in previous sections, a static modeling algorithm is discriminated

from others by differences in its strategies to integrate soft data, hard data, and

qualitative knowledge. Generally speaking, the realism of static models improves from

statistical methods to process-based methods along with the increments of qualitative

geological knowledge explicitly considered. For example, geology is quantified by

relatively loose controls in variograms or training images at the pixel level, and the

underlying hydrodynamics are modeled with equations in more realistic process-based

models (Fig. 1.1). Each algorithm has its own advantages and disadvantages, and the

choice of algorithm depends on the problem in practical cases.

Usually, there are rules of thumbs to guide the decision making for modelers,

which aim at identifying a proper algorithm for a problem. Two-point statistics is the

simplest method in the theory, implementation, and usage, thus it is still a must in all

commercial geological modeling software packages. Since the method requires a

relatively large number of wells and cannot generate complex nonlinear geometries, it

is more appropriate to model reservoirs in a relatively small domain within the

depositional basin where no complicated nonlinear geometries are detected. Multiple-

point statistics produces more complex geometries while conditioning to wells and

seismic data remains easy. Therefore, the method can model a wider domain in the

depositional system as long as training images are available. Training images

significantly increase information available for building a static reservoir model, so

multiple-point statistics does not require a large number of wells to generate satisfactory

realizations. Object-based and surface-based methods are used in problems where even

qualitative descriptions of patterns of the final deposited petrofacies are difficult to

obtain. Both methods attempt to utilize more conceptual knowledge of depositional

environments, such as geometries of petrofacies, to compensate for the lack of field

data. Object-based methods place spatial adjacency constraints on the appearance of

different types of geometries, while surface-based methods constrain placement and

6

morphology geometries with dynamic depositional rules in a forward simulation

scheme. Constraints at the scale of geometric information improves the realism of model

realizations, yet the simplicity of conditioning is sacrificed. Compared to geostatistical

methods, which are constrained at the scale of the pixel, conditioning of object-based

and surface-based model realizations is time consuming. Thus they are usually used for

cases with a very limited number of wells to condition or are used to generate training

images for multiple point statistics. Although both object-based and surface-based

methods are based on geometric information, the surface-based method aims at

imitating the sequential movement of geobodies forming stratigraphy driven by the

underlying hydrodynamic laws. The process-based methods produce very plausible

realizations by explicitly solving hydrodynamic equations. However, using

hydrodynamic equations dramatically increases the number of parameters of the

algorithm and the computational demand. Moreover, most parameters of hydrodynamic

equations are physical measures of paleofluid and paleotopographic conditions, which

are ordinarily impossible to measure or estimate in the field. Considering that hundreds

of hours are required to construct an unconditioned realization with one set of values of

parameters, a process-based method is not appropriate for the direct generation of

conditioned static reservoir models, where wide ranges of parameter values must be

explored for a Monte Carlo uncertainty estimation. As a matter of fact, process-based

methods are valuable references from which three dimensional geometry and the

relationship between the physical parameters and geometries of petrofacies are

measurable. It also improves geological knowledge on depositional environments that

are limited in real analogues, such as deepwater systems.

7

Figure 1.1: Various static reservoir modeling techniques. Along with the improvement of geological

realism from two-point statistics to process-based method, the difficulty of conditioning models

increases. After Bertoncello 2011.

Besides choosing a single modeling algorithm, a hybrid algorithm is developed to

make use of advantages of various methods with application to modeling of distributary

channel-lobe systems (Michael et al. 2010). Essentially, the hybrid modeling algorithm

belongs to the surface-based category, in which each surface is generated with the

object-based method and multiple-point statistics, demonstrating that the surface-based

method is a powerful and flexible framework to assemble surfaces generated by various

techniques. Fig. 1.2 shows the basic methodology of the hybrid algorithm. Since the

process-based method is computationally intensive, only a limited number of

realizations are obtained within the period of a modeling project. Thus, a single

realization of the process-based method is treated as a knowledge database, from which

geometric parametric parameters, e.g. length and width of lobes, are interpreted.

Probability distribution functions of interpreted geometric parameters are input to the

algorithm. The algorithm starts from sampling a length and a width from input statistics,

with which a lobe is generated in the manner of geobodies and placed in the modeling

8

grid with stochastic rules based on qualitative geological knowledge. The realization

with a single geobody is then input to multiple-point statistics as the training image to

generate a lobe conditioned to soft and hard data while the lobe length and width of the

training image remain the same. The output lobe of multiple-point statistics becomes

one surface for the surface-based framework. Iterating the steps generate a series of

surfaces, in which a series of lobe objects are conditioned to wells. Finally, a three-

dimensional stratigraphic model is obtained by stacking the surface sequence.

Figure 1.2: A hybrid modeling algorithm. Different algorithms are used at proper stages to maximize their

advantages. After Michael et al. 2010.

1.3 Surface-Based Modeling and Challenges

From the perspective of pursuing an optimal balance between geological realism

and the algorithmic simplicity, the hybrid algorithm is a promising demonstration of the

future of static reservoir modeling. The surface-based method plays a crucial role in this

framework, which is the key to combining different algorithms. First, the surface-based

method fills the gap between the underlying physics and a static reservoir model. As

noted above, the process-based method attempts to describe the depositional process by

means of hydrodynamic laws. Thus, parameters of the process-based method are

physical quantities such as flow discharges, sediment grain composition, or compaction

of flow beds, etc. Yet those parameters are not the interest of petroleum geologists and

reservoir engineers. Instead of physical parameters, which are impossible to measure

for any specific reservoirs, geometric and petrophysical parameters are meaningful in

static modeling because they are the possible forms of information from wells, seismic

9

and geological surveys. Conventional statistical methods, including two-point, multiple-

point, and object-based, describe the problem in a manner favorable to reservoir

engineering. Parameters of these methods are generally the chosen statistics of either

petrophysical properties or the geometries of petrofacies, which is straightforward and

simple. However, none of these methods are forward models; therefore, some process-

related features sensitive to reservoir production, such as erosion on the thin flow barrier

layers (Li 2008; Li and Caers 2011), have to be modeled in additional steps. The surface-

based method fills the gap in the sense of using forward simulation with geometric

parameters. Second, the surface-based method is a flexible framework to integrate

different techniques. As long as a sequence of surfaces are generated and each surface

is the function of previous surfaces, any technique can be used to generate the surfaces.

In the hybrid modeling example in Section 1.2, multiple-point statistics is chosen to

generate a surface since it focuses on improving the performance in conditioning

realizations to wells. If placement and in-situ morphology of geometries are controlled

by more complex depositional rules based on a conceptual depositional model, the

algorithm can also be applied to a statistical hypothesis test on the conceptual model

with additional real data, and thus can improve geological understandings (McHargue

et al. 2011; Sylvester, Pirmez, and Cantelli 2011). For cases where the conceptual

depositional model of geobodies are evaluated to be reliable, realizations of surface-

based models can also take the place of process-based models to provide training images

in the hybrid modeling algorithm.

Determining the placement and in-situ morphology of geometry with rules from

conceptual depositional models creates a new and unique use for the surface-based

method, namely the capability of performing a sensitivity study on the parameters of

the geological process in reservoir performance. More specifically, a new concept, the

geological dynamic process, is proposed in this dissertation, which is a valuable

direction for surface-based methods as well. Analogous to the forward dynamic process

of hydrodynamic flows used in the process-based method, a geological dynamic process

treats the spatial distribution of geometries as a forward and dynamic process. Assuming

10

a stochastic relationship exists between the location and in-situ morphology of a new

geometric body and the intermediate topography, the geological dynamic process is

described by the stochastic model. Distinguished from studying the physical process at

the scale of hydrodynamic measurements, the geological process studies the dynamics

at the scale of interpreted geometric bodies. A geological process is essentially driven

by the hydrodynamic process; however, the randomness takes a more significant role

than at the scale of hydrodynamics. The randomness is first due to events beyond the

scope of hydrodynamics, such as the occurrence of avulsion, the number and volume of

sediment events etc. The interpretational uncertainty based on experts’ judgments is the

second source of randomness, e.g. the shape measures of channel, lobe, bar, etc.

Since conceptual models in geology, including the spatial distribution model of

petrofacies (training images) and the dynamic model of depositional process

(depositional rules), are the primary source of uncertainty in estimating the reservoir

performance, the strategy of modeling a geological dynamic process should be guided

by the context of uncertainty estimation and history matching. In a new proposed

strategy known as rapid updating of geological models (Caers 2013), the prior

geological model, such as training images in multiple-point statistics, will be updated

while new wells are inconsistent with the original interpretation. If a surface-based

model is applied in the rapid updating, values of parameters controlling the depositional

rules will be updated analogously to updating training images in multiple point statistics.

Using the lobate environment as an example, parameters controlling the intensity of

stacking patterns of lobes should be updated consistently to the additional data. More

specifically, the new interpretation of the lobe stacking pattern may be intensely

stacking rather than randomly distributed as in the original interpretation. Hence, the

design of a model should provide parameters for directly controlling lobe migration,

such that the lobe stacking pattern can vary from randomly distributed to intensely

clustered simply by varying values of these parameters. The model design is also

supposed to be implemented with the simplest techniques possible, such that the

uncertainty estimation for a simple model is easier with the Monte Carlo method.

11

In current surface-based methods, the geological process relies on conceptual

depositional models. Examples of conceptual rules include trend maps of appearance

(Li and Caers 2011; Bertoncello et al. 2013), migration rates in response to slope

(Sylvester, Pirmez, and Cantelli 2011), or frequency of aggradation (McHargue et al.

2011). Conceptual rules can generate realistic stratigraphy; however, the treatment is

not ideal for uncertainty estimation, which demands more precise controls. The demand

of a more precise method to implement a depositional dynamic process in surface-based

models led to the work in this thesis. The primary challenge to implement a precise

depositional process is the lack of data. First, records of intermediate topography are

required to understand the relationships between intermediate topographies and

geometries. However, a large fraction of intermediate topographies have been removed

by sediment events, since the erosional process occurs along with the depositional

process such that intermediate topographies are not maintained in outcrops in fields.

Second, to calibrate a precise algorithm for the dynamic process of geometric bodies

would require a sequence of sufficient amounts of two-dimensional or three-

dimensional geometric bodies, such that proper parameters and coefficients can be

determined for the model. However, sufficient data from field surveys are always a

luxury. In this dissertation, a new type of data, geomorphological experiments, are

proposed as a knowledge database for the surface-base method, and a quantitative

solution is devised to select proper information for a specific real scale depositional

system from a few optional geomorphic experiments. Quantitative characterizations and

algorithms for a set of geological concepts, such as lobe stacking patterns, lobe

hierarchies are introduced, which are not only the basis of building a precise lobe

migration model but also useful tools for quantitative studies between lobate

experiments for experimental stratigraphers. Finally, a surface-based model for the

lobate environment is designed, in which the hierarchy of lobes in a realization honors

both the hierarchy of an input sequence of lobes and the conceptual compensational

stacking rule. We also introduce a measure to quantify the dissimilarity between lobe

hierarchies and demonstrate that hierarchies of lobes in realizations are statistically

12

similar to the input lobe stacking pattern, which is an effective strategy in the application

of uncertainty estimations and history matching.

1.4 The Proposed Approach

Comprehensively controlled small systems, including process-based models and

geomorphic experiments, are proposed as a feasible source of obtaining the intermediate

topographic information, because measurements can be taken at any spatial and

temporal resolution as long as the equipment permits. The geomorphic experiment was

chosen to extract erosion rules for surface-based models in this dissertation because

more complicated and diversified processes are observed in geomorphic experiments

than in process-based models. An experiment of a delta basin, performed in a tank of

five meters by five meters, was chosen, in which coarse and fine sediments with flows

were injected from a corner to form the delta fan. The intermediate topographic

information was recorded in two forms, overhead photos providing two-dimensional

information and elevations along three lines perpendicular to the injecting flow

orientation. Sequential patterns of erosion and deposition were extracted from the

intermediate topographies. A surface-based model of lobes and distributary channels

were constructed, in which each depositional event included a negative surface

representing erosion and a positive surface representing deposition. Since this

geomorphic experiment was not designed to be an analog to any real scale depositional

system and length measurements in the experiment were different from real scale

systems, the distribution functions of dimensionless ratios of erosion depth versus

deposition thickness were measured to control the relation between negative and

positive surfaces. The depositional rules were extracted and implemented in the form of

conventional conceptual models.

Models in applications of a static reservoir modeling are always required to be

analogous to the real system of a reservoir. Yet in conventional applications of

experimental stratigraphy, experimental data are usually used to improve understanding

13

of general sedimentological knowledge, and no method has been developed to specify

an experiment to a real depositional system. Specifying an experiment to a real system

by hydrodynamic laws is infeasible because physical quantities of paleoflow are not

available for any reservoir. Additionally, conventional stratigraphic methods, based on

petrophysical properties, also do not work because sediments in experimental systems

are usually simplified to two or three categories (e.g. fine and coarse) and not

comparable to the real system. Assuming that patterns of sediments in a small system

can be similar to that in a large system, defined as statistical external similarity (Paola

et al. 2009), a solution is provided to quantify similarities between sediment lobate

stacking patterns in the experiment and in the real system, so one experiment that is the

most similar to a specific real system is identifiable from a few points. The identified

experiment can be used to calibrate algorithms applicable to modeling the given real

system. The proposed solution quantifies stacking patterns of lobes with cumulative

distribution functions of pairwise lobe proximity measurements, and the similarity

between lobe stacking patterns is estimated with bootstrap two-sample hypothesis tests

based on a L1-norm difference measure between two cumulative distribution functions.

Since lobate bodies in experiments can be identified hierarchically from small scales to

large scales, depending on the decision of interpreter, the solution also includes a

treatment to quantify lobe hierarchies and to choose the most appropriate scale at which

the interpretation should be taken. This solution is demonstrated by comparing two delta

fan experiments to two published real systems, and the method successfully identified

lobe patterns from one experiment at scales with significantly high similarity to a real

system.

Once a lobe pattern is identified, it is used as input to the surface-based model.

Another surface-based model is developed, in which a new type of lobe migration

mechanism is implemented based on a correlated random walk (CRW). When used as

an input, the lobe pattern is treated as a sequence. The input migration sequence is

characterized by cumulative distribution functions of migration distance and migration

orientation, shifting the angle to control the random walk behavior of the lobe migration.

14

The simple lobe migration mechanism is controlled only by three distribution functions

from the input pattern and two coefficients, which is proper for uncertainty estimations.

A quantitative measure of the hierarchical similarity between two lobe patterns was

proposed to estimate the reproduction of lobe hierarchy in model realizations. It was

proven that lobe hierarchies are implicitly controlled in the new mechanism and all

realizations are hierarchically similar to the inputs.

1.5 Thesis Outline

This thesis consists of five chapters. The method of extracting erosion-deposition

patterns from experimental intermediate topographic information is demonstrated in

Chapter 2. In this chapter, steps of building a surface-based model based on extracted

statistics of dimensionless erosion-deposition geometric information are also

introduced.

Chapter 3 explains the methodology of identifying a proper experiment and the

proper scale of interpretation in this experiment. First, an automatic algorithm to select

lobe patterns versus scale is developed based on agglomerative hierarchical clustering.

Second, a two-sample bootstrap hypothesis test based on the L1-norm measure of

dissimilarity between distribution functions is devised to test the similarity between lobe

patterns, which is tailored to small size samples such as lobes. Finally, the method is

applied to real datasets of two experimental and two real lobe systems.

Chapter 4 demonstrates a simple strategy of modeling the lobe migration

mechanism based on a CRW that is controlled by statistics of the input lobe pattern. The

hierarchical similarity measure based on cophenetic distance in the hierarchical

clustering result is proposed to validate reproduction of lobe hierarchies in realizations

of the new model.

Chapter 5 discusses contributions of this dissertation and suggests potential future

research.

15

Chapter 2

EROSION RULES IN SURFACE-BASED

MODELING: GEOMORPHIC

EXPERIMENTS AS THE KNOWLEDGE

DATABASE

2.1 Introduction

2.1.1 Model Setting: Deepwater Turbidite System

The model in this chapter is set to be distributary channels and lobes in the lower

fans of a deepwater turbidite system. Deepwater turbidite systems transport sediments

from the edge of a shelf through the ocean slope down to the ocean basin. Based on

differences of the depositional processes and the resulting stratigraphy, a turbidite

system is divided into three sections: upper, middle, and lower fans (Fig. 2.1) from the

shelf toward the basin. The upper section features submarine canyons, where sediment

events are initiated by a ‘landslide’ on the edge of shelf. The inertial energy of sediments

keep increasing due to gravitational acceleration. While additional sediments are

16

entrained from the bed into the turbidity flow, additional clear water is entrained from

the ambient water as well. The intensity of entrainment increases along with the flow

energy. Since the flow energy keeps increasing due to the high slope in submarine

canyons, this section is dominated by erosion. The middle section starts from the toe of

the ocean slope, where accelerated gravitational flow hits the topographic surface with

a high gradient in elevation. This section includes well developed channels filled with

coarse-grained sediments. In the lower fan section, the turbidity flow weakens until its

death due to the low slope. The primary feature of the lower fan is the hierarchical

structure of lobes and distributary channels (Fig. 2.2). Lobes include relatively large

sand bodies along with shale margins. Along with the thin, interlobe, and shale layers,

hydrocarbon reservoirs may exist in the lower fan. Thus the lower fan is of interest to

the petroleum industry (Deptuck et al. 2008; Droz et al. 2003; Garland et al. 1999;

Gervais et al. 2006; Groenenberg et al. 2010; A. H. Saller et al. 2004; Sullivan et al.

2000; Van Wagoner et al. 2013). Due to the limited understanding of lobe spatial

distributions and the architecture of stratigraphy in a deepwater turbidite system, an

uncertainty estimation with static reservoir models is an important step at the appraisal

stage.

Figure 2.1: The demonstration of a deepwater turbidite system. The model setting of this work is in the

lower fan section of this system (red square).

17

Figure 2.2: A demonstration of the hierarchical structure of lobate systems. The nomenclature follows

Prélat et al. 2010.

2.1.2 Erosion in Static Reservoir Modeling

The interlobe fine grained layer (Fig. 2.3) formed by either allogenic or autogenic

processes in the lobate environment is a significant feature of deepwater systems. Fine-

grained layers are observed interbeded with sand bodies of lobes in the stratigraphy.

Since the fine grain layers are normally assumed impermeable and laterally continuous

all over the basin area, the existence of a shale layer between two lobe sand bodies works

as a flow barrier in hydrocarbon reservoirs. The thin and fine grain layers may be eroded

by successive sediment events, forming holes on the thin flow barriers, and causing

direct contact between the sand bodies of two lobes. Therefore erosion may significantly

change the production performance of a reservoir (Fig. 2.4).

Modeling erosion on fine grain layers is challenging. Erosion is conceptually

modeled as holes in the subseismic scale fine-grained layers. The spatial distribution of

eroded holes is a complicated nonlinear function of both successive sediment events and

intermediate topographies. However, obtaining sufficient data on paleoflow and

18

paleointermediate topography from field studies or seismic surveys is difficult. First,

only limited vertical two dimensional data of erosion are available in outcrops, which is

insufficient to build three dimensional erosion geometry and three dimensional

geobodies. Second, the subseismic scale fine-grained layers, ranging from tens of

centimeters to several meters, are not detected by seismic surveys; therefore, it is

impossible to quantitatively characterize the spatial distribution of eroded holes from

the seismics. Finally, the intermediate paleotopography is not maintained in any

outcrops; therefore, the relationship between spatial distribution and erosion cannot be

quantitatively characterized. Since the measurement of intermediate topographies in

active real depositional systems is also infeasible, a new source of information, where

intermediate topographies are easily available, must be used. One possible source is the

process-based model. Intermediate topographies can be recorded at any temporal

interval of the process-based model, so erosion geometry and spatial distribution of

eroded holes can be quantitatively estimated. However, the geomorphic experiment is

used in this dissertation because more complicated patterns of sediments are observed

in geomorphic experiments than in process-based models.

Using the surface-based method is appropriate for modeling spatial distribution

of erosion on fine-grained layers. Compared to conventional statistical methods, the

surface-based method is a forward model, which is naturally proper to emulate the

sequential forward process that distributes lobes in the basin. Since the surface-based

model works at the scale of the interpreted geometry of petrofacies, complex

nonstationary stratigraphy can be easily generated and the uncertainty of geometric

parameters can more easily be reduced than the physical parameters in process-based

models. Moreover, proper conceptual depositional rules in surface-based models are

computationally efficient, while the realism of model realizations is as plausible as

process-based methods.

The surface-based method still requires understanding of three-dimensional

erosion geometry and intermediate topographies, which are characterized using the

19

geomorphic experimental data in this chapter (Fig. 2.5). Generally speaking,

geomorphic experiments are simplified, small depositional systems controlled and

operated by experimental stratigraphers. Since the experiments are comprehensively

controlled, the initial conditions and the intermediate states of the processes are

recorded; therefore, the geometry of erosion can be calculated. A method of using

intermediate topographic information from a geomorphic experiment was developed.

Here, the overall workflow of surface-based modeling followed with a detailed

description of building the model is introduced. A geomorphic experiment of a delta fan

system, along with a treatment of visualizing and characterizing erosion and deposition

geometry, are also introduced. The erosion geometric information is measured with

dimensionless ratios so it is rescalable to a real system. Erosion in the surface-based

model is controlled by statistics of the dimensionless ratios.

Figure 2.3: The repetitive appearance of fine grained interlobe layers. After Prélat, Hodgson, and Flint

2009.

20

Figure 2.4: The eroded shale drapes from an outcrop. After Alpak, Barton, and Castineira 2011.

Figure 2.5: The proposed strategy of using geomorphic experiment data in surface-based models.

Experimental data are used to extract geometry and depositional rules in Step 4. Steps 1-3 refers to

Bertoncello 2011.

21

2.2 Methodology

In this section, a general workflow of a surface-based model (Section 2.2.1) and

the details of generating geometry and making the depositional rules (Section 2.2.2) are

presented. Finally, a treatment of characterizing erosion and deposition rules for the

surface-based model using delta fan geomorphic experiment data is presented. The

simulation results demonstrate that the surface-based model, coupled with proper

geomorphic experiment data, is a feasible direction for improving the surface-based

method (Section 2.2.3).

2.2.1 Surface-Based Modeling

As noted in Section 1.2, the basic idea of surface-based modeling is to imitate

with conceptual rules the geological depositional process driven by the underlying

physical laws of the turbidity current. The geological depositional process is defined as

the forward dynamic process driving the spatial distribution of geobodies. Different

from the process-based model through the perspective of physics, such as turbidity

current and sediment grains etc., the geological dynamic process considers the problem

through geobodies interpreted by geologists from petrofacies. The term ‘geobody’ refers

to lobes and channels in the lobate environment, which are at a higher spatial and

temporal scale than flow and particle. A geobody is the interpreted geometry of

sediments related to regular geometry formed by flow and sediment; therefore it reflects

underlying physics as well. Depending on the depositional environments and the scale

of interest, the geobody can represent any geometry within the hierarchy. In the case of

this dissertation, the lobe is the level of our interest in the hierarchy of lobate geometries.

Because uncertainties are involved with the interpretation of geometries and rules,

depositional rules should reflect the general physical laws and account for uncertainties

of rules as well. Therefore, the design of geological processes should be stochastic rather

than deterministic. They are normally obtained from conceptual geological

understandings about depositional processes. In most surface-based modeling

algorithms, only conceptual depositional rules are implemented because of the lack of

22

precise quantitative measurements on the intermediate conditions of the geological

dynamic processes.

Surface-based models have been developed for different depositional

environments (Bertoncello et al. 2013; McHargue et al. 2011; Pyrcz, Catuneanu, and

Deutsch 2005; Pyrcz et al. 2012; Pyrcz and Strebelle 2006). Generally, the model is

constructed based on a sequential forward workflow (Fig. 2.6). In the first step, a

template of geometric boundary is designed to represent the general shape in the

depositional system to be modeled. The template boundary is designed to be controlled

by parameters with geological interpretation, such as lobe length or width. Values of

geometric parameters of a template boundary are controlled by input statistics at each

step of the simulation. The morphology of every geobody varies based on depositional

rules. In current surface-based models, the morphology of geobodies is controlled by

centerline key points. A key point is a geologically or geometrically meaningful point

on the centerline of a template boundary, such as the most proximal point of a lobe or a

point on a channel centerline. Locations of key points are determined based on the

conceptual depositional rules such as compensational stacking. Key points determine a

centerline, and a boundary is determined by fitting the template boundary along a

centerline. Once a boundary is obtained, a property trend map is generated over the

geobody, representing the conceptual model of petrofacies within a geobody, such as

thickness, petrofacies, or petrophysical properties. With the trend map and a defined

trend-thickness function, the geobody thickness map is calculated. The current

topography is updated by adding the latest geobody thickness map on the current

topography, which is then used as topography for the next step. This procedure is

repeated until a predefined criterion, such as a top surface or a volume from the seismic

survey, is satisfied.

23

Figure 2.6: The general forward workflow for surface-based models. After Caers, 2012.

24

2.2.2 Building a Distributary Channel-Lobe Systems

A geobody is generated through a three-step procedure including template

boundary, trend maps, and geobody surfaces. The modeler determines the geometry of

geobodies and the depositional rules through close collaborations with considerations

on algorithm complexity, and purposes of the model. The geometry should be

algorithmically flexible to be adjusted according to intermediate topographies.

Template Boundary

A template boundary is defined as the edge of a geobody in the horizontal plain.

Based on overhead photos recorded in the experiments, ovals represent the general

boundary of the lobe. The distal half of a lobe is slightly wider than the proximal. The

distributary channel is interpreted as a constant width and with light sinuosity (Fig. 2.7).

Four centerline control points, all of which are geologically interpretable, are used to

vary the in-situ morphology of a geobody based on intermediate topographies. The

control points are the channel source, the channel turning, the most proximal point, and

the lobe distal point (Fig. 2.7).

The control points are defined as follows.

The channel source point is defined as the most upstream point of a geobody.

Conceptually, the sediment flow starts from the channel source for a geobody.

The lobe most proximal point (MPP) is defined as the transition point from the

channel part to the lobe part, which is the most important point in this model.

The channel turning point is defined as the location where maximum curvature

is present. Since distributary channels in the experiments show low sinuosity,

only one turning point is set on the centerline between the channel source and

the MPP.

25

The lobe distal point is defined as the most downstream point on the centerline

of a geobody.

Locations of the centerline control points are determined by depositional rules

with a geologically reasonable order (details in the following paragraphs), and the

centerline is calculated by fitting a spline curve on the control points.

Figure 2.7: The template boundary of a distributary channel-lobe geobody interpreted from overhead

photos. The channel is interpreted to be constant width and the lobe is interpreted to be an oval. Four

centerline control points with geometric or geological interpretation are defined to determine the

morphology of a specific geobody.

The template boundary is calculated along the centerline for the channel part and

the lobe part, respectively. Since the centerline morphology will be varied by

depositional rules for every specific geobody, geometric equations of the template

boundary are designed to be sufficiently robust, such that the boundary points can be

calculated along a curved centerline. The lobe boundary is defined by an oval shape

equation. Given an arbitrary lobe centerline point 𝑃𝑐(𝑥𝑐 , 𝑦𝑐) apart from MPP(𝑥0, 𝑦0)

toward the downstream direction at a normalized distance dP along the centerline, a

related boundary point 𝑃𝑏(𝑥𝑏 , 𝑦𝑏) is either of the two points whose perpendicular

distance to 𝑃𝑐 is defined by Eq. (2.1). The perpendicular distance from channel

26

boundary points to channel centerline points are set to be constantly equal to the lobe

width at MPP (𝑑𝑃 = 0).

𝑊𝑃

2= 𝐿 ∗ √

(𝒇𝑾)𝟐∗𝑨

4𝐵 ( 2.1 )

𝐴 = 𝟏 − [𝒅𝑷̅̅ ̅̅ +

𝒄𝟐−𝟏

𝟐𝟏

𝟐(𝟏+𝒄𝟐)

]

𝟐

( 2.2 )

𝐵 = 𝒆𝒄𝟏[𝒅𝑷̅̅ ̅̅ +

𝒄𝟐−𝟏

𝟐] (2.3 )

where 𝐿 is the input lobe length, 𝑓𝑊 =𝑊𝑚𝑎𝑥

𝐿 is the ratio of lobe width versus lobe length;

𝑑𝑃 =√(𝑥𝑃−𝑥0)

2+(𝑦𝑃−𝑦0)2

𝐿 is the normalized distance from point 𝑃 to MPP, ranges from

0 at MPP to 1 at Lobe End; 𝑐1 and 𝑐2 are built-in constants determining the oval

geometry.

Figure 2.8: Parameters controlling template boundary for a lobe.

Trend Maps

Besides the two-dimensional boundary, other location-related features of a

geobody, such as the spatial distribution of thickness and petrophysical facies, are

conceptualized to be functions of the geometry. Trend maps generated from geometry

describe conceptual models of spatial distribution. Two trend maps are generated

respectively for the lobe and the channel. Since properties of a channel, such as deposit

27

thickness, generally decrease from the centerline to the boundary, the channel trend map

is generated by calculating the normalized perpendicular distance from the centerline to

the channel boundary (Fig. 2.9). Lobe thickness varies gradually from MPP to the

boundary; therefore, the lobe trend map is generated by calculating the normalized

distance from MPP to the lobe boundary (Fig. 2.10).

Figure 2.9: The channel trend map generated by normalized perpendicular distance from centerline to the

boundary.

Figure 2.10: The lobe trend map generated by normalized distance from MPP to the lobe boundary.

28

Geobody Surface

For modeling erosion of lobes in surface-based methods, the concept of positive

and negative surface pairs is introduced (Fig. 2.11). If deposition is the only process

considered, a new surface is generated by adding a thickness map of the geobody onto

previous topography, so the increment of the surface is equal to the thickness of the

latest lobe. However, when erosion is considered, sediments may be removed as well as

deposited; therefore, a fraction of the new geobody will be below the previous

topography and the increment of the topography is less than the lobe thickness. In the

concept of positive-negative surface, the portion of a new lobe below the previous

topography is modeled as a negative surface and the portion above the previous

topography is the positive surface. The net-lobe thickness is calculated as the summation

of the absolute values of the negative surface and the positive surface. The surfaces are

generated with the same trend map and a trend-to-thickness function. The negative

surface depth is correlated to the positive thickness with a dimensionless ratio. Two

examples of lobe surfaces are demonstrated in Fig. 2.12.

Figure 2.11: The concept of positive-negative surfaces for erosion in surface-based models: a) a new lobe

is generated and placed onto the previous topographic surface; b) due to erosion, a portion of the new

lobe is below the previous topography; c) positive surface: the portion of the new lobe above the previous

topography; negative surface: the portion of the new lobe below the previous topography.

29

Figure 2.12: Two pairs of positive-negative surfaces. The dimensionless ratio of maximum erosion depth

versus maximum deposition thickness is 3

7.

The thickness of the positive surface and the depth of the negative surface are

calculated by two sets of trend-to-thickness functions. In Fig. 2.13, a and b demonstrate

parameters to calculate channel and lobe thickness maps. Thickness of the positive

surface of the channel is computed in Eq. (2.4).

ℎ𝑃 = ℎ1𝑚𝑎𝑥 ∗ 𝑒−

𝑑𝑃2

2𝑠2 ( 2.4 )

where ℎ𝑃 is the thickness at an arbitrary point 𝑃 within the channel; 𝑑𝑃 is the

normalized perpendicular distance from 𝑃 to the channel centerline obtained from the

channel trend map; ℎ1𝑚𝑎𝑥 is the maximum thickness of the channel positive surfaces

from input; and 𝑠 controls the decreasing rate of thickness from the centerline to the

boundary. The negative surface can be generated from the positive surface by replacing

ℎ1𝑚𝑎𝑥 with ℎ2𝑚𝑎𝑥 = ℎ1𝑚𝑎𝑥 ∗ 𝑓𝑝2𝑛 , where 𝑓𝑝2𝑛 is a negative ratio of the channel

positive thickness versus the channel negative depth. The lobe surfaces are generated

by Eq. (2.5) – (2.8).

30

ℎ1𝑃 = ℎ1𝑚𝑎𝑥 ∗ 𝑒𝐶 ( 2.5 )

𝐶 =−(𝑑𝑃−𝑑1𝐷𝑒𝑝𝑜)

2

2𝑠2∗𝑓𝑠𝑘𝑒𝑤2 ( 2.6 )

𝑓𝑠𝑘𝑒𝑤 = √−𝑑1𝐷𝑒𝑝𝑜

2

2𝑠2∗𝐷 ( 2.7 )

𝐷 = 𝑙𝑜𝑔 (ℎ𝑐ℎ𝑎𝑛𝑛𝑒𝑙

ℎ1𝑚𝑎𝑥) ( 2.8 )

where ℎ1𝑃 is the thickness in the positive surface at an arbitrary point 𝑃 within the lobe

area; 𝑑𝑃 is the normalized distance obtained from the lobe trend map; ℎ1𝑚𝑎𝑥 is the

maximum thickness of the lobe positive surfaces from inputs; 𝑠 is decreasing ratio from

the maximum thickness point to zero on the lobe boundary; ℎ𝑐ℎ𝑎𝑛𝑛𝑒𝑙 is the channel

thickness; and 𝑑1𝐷𝑒𝑝𝑜 is the distance from MPP to the thickest point of the lobe positive

surface along the lobe centerline. The negative lobe surface is computed by multiplying

a correlation ratio on the positive lobe surface.

Figure 2.13: Parameters to calculate positive-negative surfaces for a) channel and b) lobe using trend map

and trend-to-thickness functions in Eq. (2.5) – (2.8).

31

Depositional Rules

The conceptual depositional rules used in this chapter include 1) the order to

determining locations of the centerline control points of the geobody, and 2) specific

rules to determine every control point. At every step, locations of the four centerline

control points should be identified based on the intermediate topography and conceptual

rules for each points. The determination of locations of the centerline control points

follows the order 1) channel source 2) MPP 3) channel turning point 4) lobe distal point

(Fig. 2.14). This order attempts to mimic the physical reasonable behavior of a sediment

event. The placement of a new geobody starts by finding the location of the channel

source point, which is the initiation point of a sediment event entering the modeling

domain. Since behaviors of geobodies within a distributary channel-lobe system are

normally featured by the occurrence locations of lobes, the MPP is the second point to

be determined in the model. The vector 𝐴𝐶̅̅ ̅̅ (Fig. 2.14) represents the forward direction

of a sediment event, thus it is conceptually interpreted as the overall orientation of flows

forming this geobody. Because the channel turning point reflects random perturbations

around the overall orientation, it is statistically simulated within a preset range

surrounding the midpoint of segment 𝐴𝐶̅̅ ̅̅ . Specific rules for every control point are

presented as follows:

1) Channel source: the channel source is assumed to be uniformly distributed

within a segment located on the edge of the modeling grid (Fig. 2.15).

2) MPP: The new MPP location is determined using an artificial two-dimensional

probability distribution covering the whole modeling grid. The two-dimensional

distribution consists of three conceptual components. The first component accounts for

the conceptual understanding that a lobe is more likely to occur in the same vicinity as

the basin source (Fig. 2.16 a). Once a channel source is determined, distance from every

cell in the grid to the source is calculated. Probabilities of the source components are

modeled as the normalized inversed distance. The second component accounts for

32

another truth that the location of a new MPP tends to be proximate to the previous MPP

(Fig. 2.16 b). Similar to the source component, probabilities of the previous MPP

component are computed with the inverted distance to the previous MPP. The last

component accounts for the impacts of compensational stacking (Fig. 2.16 c). The

compensational rule reflects the physical truth that sediments tend to be deposited in

regions with low relief. Probabilities of the compensational rules are converted using

the normalized reciprocal of the surface relief. The final artificial two-dimensional

probability distribution function is a weighted mean of the three components,

representing the relative likelihood of a pixel to be the next MPP.

3) Channel turning: The interpretation of this rule is that a new channel avulsion

point tends to be close to the previous avulsion point (Fig. 2.17). Once the channel

source and MPP are determined, a preset window is placed around the midpoint of

segment 𝐴𝐶̅̅ ̅̅ . Probabilities of cells within the window are determined by the normalized

inversed distance from each cell to the previous channel turning.

4) Lobe distal point: Since the power of the sediment flow starts to weaken from

the MPP toward the downstream direction, segment 𝐵𝐶̅̅ ̅̅ is assumed to represent the

mean direction of the lobe (Fig. 2.14). The exact lobe orientation represented by vector

𝐶𝐷⃑⃑⃑⃑ ⃑ is sampled from a predefined gaussian distribution with 𝐵𝐶⃑⃑⃑⃑ ⃑ as the mean in order to

incorporate some randomness. The coordinate of D is calculated with vector 𝐶𝐷⃑⃑⃑⃑ ⃑ and the

given lobe length 𝐿.

33

Figure 2.14: Locations of the centerline control points of a geobody are determined following the order

1) channel source 2) MPP 3) channel turning 4) lobe distal point.

Figure 2.15: The channel source is assumed to be uniformly distributed within a segment around the

midpoint of an edge of the modeling grid.

34

Figure 2.16: The depositional rules for the second control point: lobe Most Proximal Point (MPP). The

MPP is drawn from a combined conceptual two-dimensional probability distribution on the modeling

grid. The distribution is the weighted mean of three components: a) the next MPP (black dot) tends to be

close to the source (star), so for each point in the grid, the probability is proportional to the reciprocal of

its distance to a predefined basin source; b) the next MPP (black dot) tends to be close to a previous MPP

(blue dot); c) the next MPP (black dot) tends to appear in the region with fewer deposits; d) the final two-

dimensional probability map is a weighted mean of the three components.

Figure 2.17: The depositional rule for channel turning. Once the channel source A and lobe most proximal

point C are determined, a preset window is placed around the midpoint of segment 𝐴𝐶̅̅ ̅̅ . The conceptual

probability within the window is determined by the normalized reciprocal distance from the previous

channel turning G to every cell within the window. The new channel turning B is sampled within the

window.

35

2.2.3 Geomorphic Experiments

The developed workflow of extracting erosion rules are presented with an

experiment of a delta basin, the DB-03 experiment. In this section, the experiment

setting is briefly introduced. Then, the method of visualizing and characterizing erosion

geometry and patterns from the recorded intermediate topography is presented.

Additionally, the correlation between erosion depth and deposition thickness is

characterized by a dimensionless ratio, such that the experimental information is

applicable to the length scale of real systems, is proposed. The cumulative distribution

functions of dimensionless ratios are inputs to the lobate model documented in Section

2.2.2.

Experiment Setting

The DB-03 experiment was performed and documented by the Sedimentology

Group, San Anthony Falls National Laboratory, University of Minnesota. The original

purpose of this experiment was to study the creation and preservation of channel-form

sand bodies in alluvial systems (Sheets, Paola, and Kelberer 2009). The data from the

DB-03 experiment have also been used in studies on the compensational stacking of

sedimentary deposits (Straub et al. 2009) and the clustering of sand bodies in fluvial

stratigraphy (Hajek, Heller, and Sheets 2010). Only a brief introduction of the

experiment is made in this section. For details of the experiment, refer to Sheets, Paola,

and Kelberer 2009.

The DB-03 experiment was performed in a tank measuring five meters by five

meters and 0.6 meters deep (Fig. 2.18). The sediment infeed point was placed on a

corner of the tank. A prerun was performed in which a delta fan was created as the initial

topography. The initial topography was a symmetric delta with the radius of 2.5 meters

toward the downstream direction from the sediment infeed point. The tank floor was

controlled so that the shoreline of the delta was maintained at an approximately constant

36

location. The injected sediment was a mixture of silica sand (coarse) and anthracite coal

(fine).

A subaerial laser scanner recorded intermediate topographies of the delta at

intervals of every 2 minutes along three lines perpendicular to the overall infeed flow

orientation. The three measurement lines were located at 1.5 meters, 1.75 meters, and 2

meters apart from the infeed point in the downstream direction, respectively (Fig. 2.19).

The experiment lasted 30 hours and produced an average of 0.2 meters of stratigraphy.

1180 records of intermediate topographies were available at every line. A fraction of

the final stratigraphy is shown in Fig. 2.20. Erosion and deposition are identified by

comparing the differences between two successive records (Fig. 2.21). Formal

definitions of topography, stratigraphy, erosion, and deposition are presented as follows:

Topography is defined as the intermediate top surfaces at every 𝑡, represented by a

spatial-temporal function 𝑍(𝑥 , 𝑡), 𝑥 is the spatial coordinate (one dimension along

a topographic line); the temporal coordinate is a scalar 𝑖 = 1,2,3, … , 𝑡 ,

representing the sequence of records;

Stratigraphy at 𝑡 is defined as all deposited surfaces previous to 𝑡, represented by

𝑆(𝑥 , 𝑖 = 1,2, … , 𝑡 − 1);

Erosion from 𝑡 − ∆𝑡 to 𝑡 is defined by 𝐸(𝑥 , ∆𝑡) , where 𝑍(𝑥 , 𝑡) <𝑍(𝑥 , 𝑡 − ∆𝑡) ,

implying sediment removal caused by the depositional process within the interval

∆𝑡;

Deposition from 𝑡 − ∆𝑡 to 𝑡 is defined by 𝐷(𝑥 , ∆𝑡), where 𝑍(𝑥 , 𝑡 − ∆𝑡) < 𝑍(𝑥 , 𝑡),

implying additional deposits within the interval ∆𝑡;

If 𝑍(𝑥 , ∆𝑡) = 0, the topography is not modified by a sediment event during ∆𝑡;

37

The experiment was a subaerial delta but not a deepwater system; however, the

treatment is still applicable as long as a similar form of records is measured from a

subaqueous experiment.

Figure 2.18: The experiment setting for DB-03. The experiment was performed in a tank of five meters

by five meters, in which the infeed point was set at a corner of the tank. A prerun was performed to

generate the initial topography, which was a symmetric delta with the radius of 2.5 meters apart from the

infeed point.

Figure 2.19: The intermediate topography of the experiment was measured along three lines at 1.5 meters,

1.75 meters, and 2 meters apart from the infeed point, respectively.

38

Figure 2.20: A fraction of the final stratigraphy measured on one topography line, in which the

intermediate conditions are partially removed by following events.

Figure 2.21: The erosion and deposition are identified by computing differences between two successive

records. Erosion is defined by reduced elevation between two successive elevation records, while

deposition is defined by increased elevation.

39

Erosion-Deposition Patterns

For the purpose of applying experimental data to the positive-negative surfaces

scheme, illustrated in Fig. 2.11, geometries of erosion and deposition were all measured

relative to previous topography. A sediment event plot was devised to visualize the

sequential pattern of erosion and deposition geometries. In the sediment event plot,

geometries were visualized in the form of net relative elevation changes, and all

geometries were plotted along the sequence of occurrence. The plot is demonstrated

with an example of two successive records (Fig. 2.22). In the original measurement of

elevation along a line, the y-axis represents elevation. An erosion region (blue) is

identified at 𝑇 = 1 (the upper plot in Fig. 2.22 a). Then deposition (red) occurred at 𝑇 =

2 (the upper plot in Fig. 2.22 b). In the sediment event plot, changes in topography are

plotted at the same spatial location (x-axis) as in the original plot, but the geometry of

topographic changes for all records are plotted as their order of occurrence (y-axis)

(lower plot in Fig. 2.22 a and b). The sediment event plot is a quantified visualization

of 𝐸(𝑥 , ∆𝑡) and 𝐷(𝑥 , ∆𝑡). An example of the topography evolution plot of 100 records

is shown in Fig. 2.23. First, the sediment event plot reveals that erosion usually occurs

along with deposition, which is consistent with the scheme of negative-positive

surfaces.

Three categories of geometries can be observed in the sediment event plot. In the

first category, the width of erosion geometry is greater than the width of deposition,

implying the depositional process is erosion-dominated at this stage. The width of

erosion is less than the width of deposition in the second category, so the process is

deposition-dominated. In the last category, deposition occurs with no erosion. Since

three-dimensional geometries of the erosion and deposition are not available, we must

assume that different types of geobodies determine different types of geometry pairs.

Interpretations for each category are presented in Table 2.1. Since channels are erosion

dominated, the first category is interpreted to measure from the channels. Additionally,

lobes are deposition-dominated, so the second and third categories are both interpreted

40

to be lobes. However, some lobes occur with erosion (the second category) and other

lobes occur without erosion. The probability of lobes occurring with erosion is 0.29.

Measurements of erosion depth or deposition thickness are not directly applicable to

real scale systems because length measurements in experiments are different from those

in real systems. Based on the observation that erosion and deposition geometry in two

successive records are similar, a dimensionless ratio 𝑟 = ℎ𝐸𝑟𝑜𝑠𝑖𝑜𝑛

ℎ𝐷𝑒𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 characterizes the

relationship between the maximum erosion depth and the maximum deposition

thickness (Table 2.2). Probability distribution functions of 𝑟 for the channel and for the

lobe with erosion are input into the surface-based model.

41

Figure 2.22: Examples of generating a sediment event plot from intermediate topography records: a) to

identify the relative elevation change regions between two successive intermediate topography records

(the upper plot). The relative change geometry is plotted as its temporal order (the lower plot); b) to

identify the series of geometries of the elevation change and plot the all geometries along the vertical axis

as the order of occurrence.

42

Figure 2.23: The sediment event plot of 100 records on the topography line at 1.75 meters apart from the

infeed point.

Table 2.1: Three categories of patterns are interpreted from the sediment event plot 1) channels, where

the width of erosion is equal or greater than that of deposition; 2) lobe with erosion, where the width of

deposition is greater than that of erosion; 3) lobe without erosion.

43

Table 2.2: Dimensionless ratios are used to characterize the relationship between erosion-deposition

geometries identified from the sediment event plot. Two probability functions respectively for channel

and for lobe with erosion are interpreted. The probability of erosion occurs with lobes is 0.29.

2.3 Simulation Results

A test simulation of the lobate model is performed to demonstrate the application

of erosion rules with input statistics from the DB-03 experiment. As indicated earlier,

statistics from the experiment include two distribution functions of the dimensionless

ratio r for the channel and for the lobe with erosion. The model setting and deposition

rules were demonstrated in Section 2.2.2. To account for the input statistics, negative-

positive surfaces of the channel are generated with 𝑟𝑐ℎ𝑎𝑛𝑛𝑒𝑙 sampled from its probability

distribution function. Occurrence of erosion within lobes is controlled by the probability

0.29. If erosion occurs, 𝑟𝑙𝑜𝑏𝑒 is sampled to generate negative-positive surfaces for the

lobe part. If erosion does not occur, a positive surface is generated without a negative

surface. Important geometric and simulation parameters are listed in Table 2.3. It is

noted that input statistics are extracted from two-dimensional vertical sections, while

the realizations are three-dimensional. However, the treatment is still applicable if three-

dimensional intermediate topography records are available. The implication of the

current treatment is to demonstrate the capability of producing complex stratigraphy

44

and reproducing the input experimental statistics with surface-based models. The

intermediate status of the first four lobes is presented in Fig. 2.24: the first lobe is

deposited without erosion, while Lobe 2 – 4 all erode previous topography. Two cross

sections perpendicular to the basin flow orientation monitor changes in the stratigraphy

made by every event. A model realization of 40 lobes and two cross sections of

topography are shown in Fig. 2.25. Cross sections in Fig. 2.24 and 2.25 demonstrate

that the surface-based model can produce stratigraphy that is more realistic and more

complicated than conventional static reservoir modeling techniques.

A fraction of the sediment event plot from a realization of 100 lobes is compared

with that from the experiment (Fig. 2.26). The two plots are different from the

perspective of the geometry of negative-positive surfaces, which is reasonable because

no three-dimensional geometric information is available to calibrate the trend-to-

thickness function in the model. However, the similarity between the two plots is that

all three categories of negative-positive surfaces were observed in the simulation as

expected. The QQ-plot (Fig. 2.27) compares the simulated and experimental distribution

functions of 𝑟𝑐ℎ𝑎𝑛𝑛𝑒𝑙 and 𝑟𝑙𝑜𝑏𝑒 and demonstrates the reproduction of input statistics of

surface-based model. Finally, erosion occurred in 32% of lobes in the realization, which

is close to the inputs. Notice that the distribution functions of the 𝑟𝑐ℎ𝑎𝑛𝑛𝑒𝑙 and 𝑟𝑙𝑜𝑏𝑒 and

the probability of lobes with erosion are measured from the three-dimensional

realization.

Grid Dimension 300 x 250

𝑑𝑥 1

Lobe Length (𝐿) 50*𝑑𝑥 to 130*𝑑𝑥 uniformly distributed

Lobe Width 0.3*𝐿 to 0.7*𝐿 uniformly distributed

Lobe Thickness 0.002*𝐿 to 0.008*𝐿 uniformly distributed

Initial Surface Flat

Table 2.3: Primary simulation parameters.

45

Figure 2.24: The intermediate steps of the simulation for the first four lobes. Two cross sections are used

to monitor stratigraphic changes at every step. Erosion does not occur in Lobe 1 while Lobe 2 – Lobe 4

all cause erosion.

46

Figure 2.25: A realization of 30 lobes. Two cross sections along and perpendicular to the basin flow

orientation are also shown.

Figure 2.26: A fractions of the experimental and simulated sequential pattern plots. Geometry of erosion

and deposition are different, since no three-dimensional data are available to calibrate the trend-to-

thickness function. The occurrence of paired negative-positive surfaces for the channel and lobes (with

and without erosion) are observed in the simulated result, as expected.

47

Figure 2.27: The QQ plots prove the reproduction of cumulative distribution functions of the

dimensionless ratios in the model realization.

2.4 Chapter Summary

A workflow integrating intermediate topographic data measured from

geomorphic experiments was developed based on the surface-based modeling

techniques. The workflow includes a solution for extracting sequential pattern

information on erosion and deposition from intermediate topography records. The

geometric information from the experimental data was characterized by dimensionless

ratios, such that the information from small experiments is applicable to the real-scale

depositional system. A surface-based model of a distributary channel-lobe system was

constructed to demonstrate the method. It was demonstrated that using the surface-based

model could reproduce input statistics and generate complex and realistic stratigraphy.

The experiment used in this chapter only included two-dimensional intermediate

topographies; however, the workflow is also applicable to experiments with three-

dimensional intermediate topography records. The current surface-based model only

accounts for geometric information on deposition and erosion, and further work may

involve improvements on depositional rules with experimental data to consider the

spatial-temporal clusters of sediment events.

48

Chapter 3

THE STATISTICAL EXTERNAL

SIMILARITY BETWEEN LOBATE

ENVIRONMENTS: LINKING

EXPERIMENTS TO REAL-SCALE

SYSTEMS

3.1 Introduction

For the integration of experimental data into reservoir models, the first challenge

is to build the link between unscaled experiments to real scale depositional systems. In

terms of experimental stratigraphy, the similarity between a small version of a system

and a large system is defined as the external similarity (Paola et al. 2009). The external

similarity is defined distinct from internal similarity, which describes the phenomenon

of a small part similar to the whole system. Internal and external similarities are both

manifestations of scale dependence. Normally, experimental stratigraphers consider

internal similarity as the implication of external similarity. External similarity is the

49

fundament for linking an experiment to a real depositional system. At present, although

we have much to learn about the physics behind the external and internal similarity, they

are observed in experiments of various environments. Hence the detection of the

external similarity is the objective of this chapter. Unfortunately, neither quantitative

solutions to estimate the external similarity, nor methods to specify a geomorphic

experiment to a specific real system have been published. Developing a solution linking

an experiment to a real depositional system with hydrodynamic laws or conventional

stratigraphic method is complicated. From the perspective of hydrodynamics,

specifying an experiment to a real system includes setting physical quantities such as

flow discharge and sediment grains to be analogous to the unknown paleoflow of the

real system, which clearly does not work. Conventional stratigraphy compares

depositional systems through the geological history of the systems, which is interpreted

based on the spatial-temporal distribution of petrophysical properties (Deptuck et al.

2008; Droz et al. 2003; Gervais et al. 2006; Gervais et al. 2006; Jegou et al. 2008; Prélat

et al. 2010; A. H. Saller et al. 2004; A. Saller et al. 2008). However, geomorphic

experiments are extremely simplified systems, in which the sediment composition

normally contains only two or three representative types of grains, such as fine and

coarse. Thus, stratigraphers rarely consider deposits in experiments to be analogous to

a real system.

A statistical method is proposed in this chapter that defines a quantitative and

statistical measure on the external similarity between an experimental system and a real-

scale depositional system, such that an experiment that is the most similar to the real-

scale system can be identified and used as a reference. Then, the external similar

experiment is treated as a reference from which the characterized information is

applicable in understanding and simulating the real system. The basic assumption of

this chapter is that external similarities exist between lobate systems. The solution was

devised in a favorable manner for engineering applications, such that the results can be

ready to use in surface-based methods for static reservoir modeling. The devised

solution consists of two stages of analysis along with a data preprocess step (Section

50

3.2). Since the intermediate states of the depositional process are recorded by overhead

photos, the preprocessing step is required to extract lobate polygons from photos. In the

first stage of this method, a treatment to parameterize lobe stacking patterns based on

pairwise lobe proximity and an automatic algorithm to quantify experimental lobe

hierarchies are presented. The algorithm is required because lobe stacking patterns at

various scales in the experiment must be estimated for external similarity, for which

manual interpretation is infeasible. In the second stage, lobe stacking patterns are first

obtained from the quantified lobe hierarchy. Then, the lobe stacking pattern is

characterized as a spatial point process. Two point processes from respective lobe

stacking patterns are compared based on the nearest neighbor statistics. A two-sample

bootstrap hypothesis test based on L1-norm differences between cumulative distribution

functions was designed to estimate the similarity between nearest neighbor statistics

with a probability value. Each stage is demonstrated by examples followed by an

application with the whole workflow for testing the external similarity between two

experimental and two published real-scale lobe stacking patterns (Section 3.3). Finally,

results of the method are discussed on a technical level.

3.2 Methodology

The method aims to quantify the similarity between two lobe stacking patterns. If

stacking patterns from an experiment are similar to that from a real-scale system, the

experiment can be used as a database of sedimentological knowledge for the real system.

Therefore, depositional rules can be characterized and calibrated with comprehensive

experiment data not available in the real system, and the determined rules are then

applicable to model a real-scale system using surface-based methods. Since the

intermediate conditions of the experiment were recorded by overhead photos, the

preprocessing step for extracting lobes from photos is introduced in Section 3.2.1. Stage

One of the method (Section 3.2.2) included parameterization and hierarchy

quantification. A lobe stacking pattern was parameterized by the pairwise proximity of

every pair of lobes within the stacking pattern. A hierarchical clustering algorithm was

51

designed to automatically quantify the hierarchy of lobes in experiments based on the

parameterization. The lobe hierarchy is a quantitative description of the function

between scales of interpretation and lobe stacking patterns, such that lobe stacking

patterns could be extracted from the quantified lobe hierarchy at different scales. Once

a stacking pattern was obtained from the hierarchy, it was compared to the stacking

pattern interpreted from the real-scale depositional environment. Section 3.2.3 describes

a special hypothesis test based on bootstrap and L1-norm measures that was designed

to estimate a p-value for measuring similarities between the two stacking patterns.

3.2.1 Experimental Data and Preprocessing

Other than DB-03 in Chapter 2, two alternative delta fan experiments are

described in this chapter. The settings of the new experiments were similar to DB-03.

Both experiments were performed in the same tank as the DB-03, and the sediment

infeed points were set on a corner of the tank. The frequency of intermediate

topographic records in these experiments were not sufficient to estimate the lobe at fine

scales, so only overhead photos obtained at the interval of 30 seconds became available

to measure the intermediate status of the lobes in the process. The overhead photos were

recorded by a camera over the tank. Since photos were geometrically corrected to

eliminate distortions caused by the lens and location of the camera (Fig. 3.1), the shape

and length measurements in the horizontal plane of photos are accurate.

The preprocessing on overhead photos involved three tasks:

Identification of a successive fraction in the overhead photo sequence, such that

behaviors of a single active channel and its connected lobe were recorded.

Experiments used in this chapter were performed for more than 100 hours, and

complex features, such as multiple channels and lobes, were observed. Since the

surface-based model in this work only considered one lobe at each step, a very

small fraction of the photo sequence was selected, such that only one active

channel with its lobe appears in each photo.

52

Interpretation and clipping of the lobe region from a photo. Manual

interpretation was required to distinguish the lobate geometry from the complex

background of an overhead photo. Since photos were recorded at very fine

intervals, a new lobe was identified only when a significant shift was observed

on the end of the channel between two successive photos. Otherwise, only one

lobe is identified from multiple photos (Fig. 3.2).

Extraction of geometric features for lobes, including the lobe MPP, lobe

orientation, and the lobe boundary, from clipped photos. An automatic script

based on the image processing toolbox in Matlab was developed for this

purpose. The geometric features of a lobe are defined as follows.

A lobe boundary is defined by the outmost cells of the cropped lobe in

an image.

The MPP of a lobe is defined as a lobe boundary point that is the closest

to the sediment infeed point.

Lobe orientation is defined by the unit vector from the lobe MPP to the

geometric centroid.

53

Figure 3.1: Corrected overhead photos of an experiment taken at regular intervals, in which only one

active channel was recorded within the successive photos. Courtesy by Prof. Chris Paola, San Anthony

Falls National Laboratory.

Figure 3.2: The lobe connected to the latest channel was interpreted on every photo, and a series of lobes

are obtained.

54

3.2.2 Stage One – Automatic Lobe Hierarchy Quantification

As the prerequisite for a quantitative study, a parameterization strategy for lobe

stacking patterns was proposed based on the pairwise proximity between the lobes.

Since lobes are polygons with geological interpretations, the pairwise lobe proximity is

measured by a geological distance function designed to account for both geometric and

geological proximity. Based on the lobe proximity, the hierarchy of parameterized lobes

can be quantified with agglomerative hierarchical clustering. A temporal constraint was

designed for the clustering algorithm, such that the clustering algorithm followed the

sequence of lobes. The method is demonstrated with a sample sequence of lobes.

Parameterization of Lobe Stacking Patterns

Stacking patterns are differentiated by the intensity of regional lobe clustering. In

terms of spatial statistical analysis, intensity of regional lobe clustering is reflected by

the statistics of pairwise proximity between lobate polygons. The pairwise geological

proximity of lobes were measured by four parameters, namely, the lobe MPP proximity,

the lobe orientation proximity, the polygonal proximity, and the shape proximity.

MPP proximity 𝒅𝑴𝑷𝑷: The MPP of a lobe is defined as the boundary point of

a lobe that is the closest to the infeed point of an experiment tank (Fig. 3.3 a). In

the conceptual model of flow process in lobate system, the MPP is interpreted

as the starting point where the energy of the sediment flow starts weakening.

Thus, the spatial proximity between MPPs implies genetic differences between

two lobes.

Orientation proximity 𝒅𝜽: Lobe orientation is defined by the unit vector from

the lobe MPP to the geometric centroid of the polygon (Fig. 3.3 b). The lobe

orientation records the orientation at the end of the life of a sediment flow.

Similar to MPP, the lobe orientation proximity also reflects the genetic

dissimilarity between two lobes.

55

Polygonal proximity 𝒅𝒑: Although lobes are usually marked by MPPs, they are

geometrically polygons. Hence the polygonal proximity must also be studied to

measure the areal density. In this work, the polygonal proximity is measured by

the Haussdorff distance (Veltkamp 2001) (Fig. 3.3 c) because of its simplicity.

The Haussdorff distance is the maximum distance from a point set to the nearest

point in another set, defined by Eq (3.1) – (3.2). As indicated, two lobes are

represented by the boundary points while calculating the polygonal proximity.

ℎ(𝐴, 𝐵) = 𝑚𝑎𝑥𝑎∈𝐴

{𝑚𝑖𝑛𝑏∈𝐵

[𝑑(𝑎, 𝑏)]} ( 3.1 )

𝐻(𝐴, 𝐵) = 𝑚𝑎𝑥 {ℎ(𝐴, 𝐵), ℎ(𝐵, 𝐴)} ( 3.2 )

Shape proximity 𝒅𝒔: Small lobate geobodies within a larger one normally share

similar shapes. The shape proximity is measured by applying procrustes analysis

on the boundaries of two lobes (Fig. 3.3 d). In the procrustes analysis, two lobes

are first translated and rotated such that the MPP and lobe orientation are

overlapped. Then, coordinates of a lobe boundary are normalized by the Root

Mean Square Distance (RMSD) (Eq. (3.3) – (3.4)), which is a statistical measure

of the scale of the lobes. Finally, differences between lobes are calculated along

two normalized boundaries.

𝑠 = √∑ (𝑥𝑖−𝑥𝑀𝑃𝑃)2+(𝑦𝑖−𝑦𝑀𝑃𝑃)2𝑘

𝑖=1

𝑘 ( 3.3 )

𝑥�̅� =𝑥𝑖−𝑥𝑀𝑃𝑃

𝑠, 𝑦�̅� =

𝑦𝑖−𝑦𝑀𝑃𝑃

𝑠 ( 3.4 )

56

Figure 3.3: The parameterization of lobe stacking patterns: a) the source is defined as the MPP on the

lobe boundary to the basin source; b) lobe orientation is defined by the unit vector pointing from the

source point to the geometric centroid of a lobe; c) polygonal proximity is measured by the Haussdorff

distance between two lobes; d) shape proximity is measured through procrustes analysis.

Hierarchy of Lobes

Studies on real lobate depositional systems reveal a hierarchical organization of

lobate geobodies (Prélat et al. 2010). The hierarchy includes event beds, lobes, lobe

complexes, and fans, varying from thinnest and smallest to thickest and largest. In real-

scale depositional systems, determination of such a hierarchy is based on

sedimentologic and stratigraphic studies of the spatial-temporal distribution of

petrophysical properties (e. g. grain sizes, stratigraphic formations, paleoflow

conditions, and deposit thickness). Within a hierarchy of lobate geobodies, the level of

lobes and the lobe elements are of the most interest for petroleum exploration and are

the scale to which static reservoir modeling is applied. Hierarchies of lobate geobodies

exist in geomorphic experiments as well. However, in experiments, the stratigraphic

method is not able to identify a level within the hierarchy that is equivalent to lobes

within a real-scale system, since the experimental processes are extremely simplified

and upscaled concerning flow conditions, grain sizes, etc. Moreover, data recorded in

57

geomorphic experiments are extremely substantial (e.g., tens of thousands of overhead

photos recorded every 30 seconds) compared to data obtained from real systems through

geological field studies and seismic surveys (e.g., several or less than 20 lobes).

Therefore, conventional stratigraphic methods based on expert analysis is essentially

infeasible.

Based on the assumption that a statistical and external similarity exists between

small depositional systems (experimental) and large depositional systems (real scale)

(Paola et al. 2009) and the capability of characterizing a lobate system by its hierarchy

of lobes, the algorithm presented in this section aims for automatic and quantitative

characterization of lobe hierarchies in experiments based on the proximity measure

between lobes. A hierarchy of lobes is a description of the function between scales of

interpretation and lobe stacking patterns. The scale of interpretation is the scale on

which geobodies of petrofacies are interpreted. If the geological dissimilarity between

a set of lobes are below a chosen scale of interpretation, they are interpreted as an

integrated lobe at the chosen scale. The proximity measure should integrate

dissimilarities between two lobes in a spatial, temporal, and geological sense based on

the four parameters indicated earlier. For example, MPPs of small lobes forming a

higher scale lobe are located within the intermediate vicinity. Moreover, these small

lobes within the same higher scale lobe must be successive in time, since temporal gaps

lead to the interpretation of multiple lobes. Finally, small lobes of a higher scale lobe

usually share similar shapes and orientations. Once the pairwise proximity of lobes are

quantified by the four parameters, the overall hierarchy of a dataset is quantified by the

hierarchical clustering algorithm. The challenge is to constrain the general statistical

algorithm, such that the results are geologically reasonable.

58

Agglomerative Hierarchical Clustering with Geological Adjustments

The most extensively applied data-mining algorithm for summarizing data

hierarchies is agglomerative hierarchical clustering. Given a dataset 𝐺 of 𝑁, the basic

process of agglomerative hierarchical clustering is as follows (Johnson 1967).

Step 1: Compute the similarity between every pair of observations in 𝐺 , and

assign each single observation into its own cluster, defined as 𝑁 clusters.

Step 2: Identify the most similar pair of clusters and merge them into a single

cluster. Now the number of clusters is 𝑁 − 1.

Step 3: Compute the pairwise similarity between the new cluster and the old

clusters.

Step 4: Repeat Step 2 – Step 3 until 𝑁 = 1

The similarity is measured using distance, and Euclidean distance was chosen for

this work. Different strategies can be performed in Step 3 to compare the similarity

between clusters, such as single-linkage, complete-linkage, etc. In this chapter, the

single-linkage, the distance between the centroids of two clusters, is used. Hierarchical

clustering has been extensively applied as a powerful tool for data analysis. However,

adjustments reflecting stratigraphic interpretation rules must be applied on the normal

hierarchical clustering procedure to generate geologically reasonable groups of lobes.

Since the geological proximity between two lobes is measured through the four

parameters defined earlier, the first adjustment is the definition of a distance function

integrating all four parameters. Given a set of lobes 𝐺, a single lobe g is measured by a

four-dimensional vector 𝑔 = (𝑀𝑃𝑃, 𝜃, 𝑃𝑜𝑙𝑦𝑔𝑜𝑛, 𝑆ℎ𝑎𝑝𝑒) . As an alternative to the

normal treatment of clustering in the four-dimensional space, the proximity between

two lobes is defined by the weighted mean of the proximity in the four respective

59

components (Eq. (3.5)) for the purpose of accounting for expert judgment on the

importance of each parameter in the interpretation.

𝑑𝑓 = 𝑤𝑀𝑃𝑃𝑑𝑀𝑃𝑃 + 𝑤𝜃𝑑𝜃 + 𝑤𝑝𝑑𝑝 + 𝑤𝑠𝑑𝑠, ∑𝑤𝑘 = 1, 𝑘 = 𝑀𝑃𝑃, 𝜃, 𝑝, 𝑠 ( 3.5 )

where empirical weights reflect the importance of each component. In the determination

of grouping two small lobes into a higher scale lobe during the interpretation, the

component with the higher weight is considered more important. For example, one set

of weight values used in this work that give a reasonable clustering of lobes as 𝑤𝑀𝑃𝑃 =

0.4; 𝑤𝜃 = 0.2;𝑤𝑝 = 0.3;𝑤𝑠 = 0.1 , implying that the interpreter considers the

dissimilarity between MPPs as the most important criteria in the interpretation. In this

manner, proximities of four parameters are fused into a single value. In the following

sections, the fused proximity is named as the geological distance, and the one-

dimensional space of geological distance is defined as the interpretation space. Any

scale comparison in the following sections is in the interpretation space.

The second adjustment is the temporal constraint, which is a typical stratigraphic

consideration that has not been studied in the algorithmic literature on clustering. During

the interpretation procedure for a lobate depositional system, two lobes may be close

from the perspective of the four parameters but are separated by a relatively long gap

from the perspective of time. In the lobate reservoir, especially in deepwater systems,

the time gap between two spatially overlapped lobes is usually indicated by thin

impermeable shale layers. These thin layers are difficult to measure in experiments.

Hence, the purpose of a temporal constraint is to distinguish two clusters of lobes that

are spatially and geometrically close but temporally separated. Distinct from assigning

empirical weights to four proximity components, which is a ‘soft’ adjustment on the

hierarchical clustering, the temporal adjacency constraint is a ‘hard’ constraint, forcing

the agglomeration of clusters to occur only in two geobodies that are successive in time.

For example, given a lobe sequence 𝑔𝑖 , 𝑖 = 𝑡1, 𝑡2, 𝑡3, the acceptable agglomerations are

{𝑡1, 𝑡2} or {𝑡2, 𝑡3} , while {𝑡2, 𝑡3} is not allowed (Fig. 3.4 a). Instead of directly

modifying the standard hierarchical clustering algorithm, which is efficient and

60

available in most statistical toolboxes, a sequential cumulated distance (Eq. (3.6)) is

applied to ensure that the distance between two points closer in time is always closer in

the interpretation space.

𝑑(𝑖,𝑗) = ∑ 𝑑𝑓(𝑘,𝑘+1)𝑗−1𝑖 ( 3.6 )

where 𝑖, 𝑗 are temporal indices of a pair of points in the sequence; 𝑑𝑓(𝑚,𝑛) is defined in

Eq. (3.5). The essence of the sequential, cumulated distance between the ith point and

jth point in time is the sum of segments linking each successive pair of points between

i and j, which ensures that 𝑑(𝑖,𝑗) is always greater than 𝑑(𝑖,𝑘), 𝑖 < 𝑘 < 𝑗 , such that

agglomeration only occurs in temporally successive pairs of clusters (Fig. 3.4 b).

The results of hierarchical clustering can be visualized by a dendrogram (Fig. 3.5),

which is a weighted binary tree structure. Each leaf of the tree represents an observation

(a lobe) from the clustered dataset, and each branch of the tree structure represents a

cluster, or a higher-scale lobe. The height of the junction of two branches represents the

distance between two clusters. Thus, a dendrogram describes the hierarchy of lobes as

a function between the scale of interpretation and the configuration of clusters.

The lobe hierarchy described by a dendrogram is straightforward; however, the

treatment works well only with small datasets. In cases with hundreds of observations

and multiple levels in the hierarchy, visualizing lower levels of the tree structure

becomes infeasible (Fig. 3.6). Thus, the reachability plot (Fig. 3.7), an alternative form

of dendrogram, is introduced for large datasets.

The reachability plot (Ankerst et al. 1999; Breunig et al. 1999; Sander et al. 2003)

is a bar chart representing clustering results of density-based clustering algorithms.

Studies have proved that a reachability plot is equivalent to a dendrogram in certain

cases and can be directly converted from dendrograms resulting from hierarchical

clustering. Each observation in the clustered dataset is represented by a bar in the

reachability plot, and bars are aligned along the horizontal axis, maintaining the original

61

order of leaves of the dendrogram. The height of a bar is the distance from an

observation to all observations on its left, which is the height of the common junction

between a leaf and all leafs on its left in the dendrogram. Tall bars in a reachability plot

represent large intercluster gaps and short bars represent short gaps. Thus, valleys in a

reachability plot represent a dense set of lobes. The advantage of a reachability plot is

that the method is able to properly visualize large datasets with hundreds of

observations.

The interesting property of a reachability plot, or a dendrogram, is that the y-axis

represents scales of interpretation, and valleys in bars represent a configuration of

clustering. In other words, the dendrogram defines a function between scales of

interpretation and the stacking pattern of lobes. A scale of interpretation is determined

by thresholding the y-axis. Bars greater than the chosen scale intersect with the

threshold, and a valley between two intersections is a cluster. Each cluster is then a lobe

at the chosen scale of interpretation (Fig. 3.8). If the y-axis of a reachability plot is

thresholded from the finest to the coarsest spatial and temporal scales recorded in an

experiment, the respective stacking pattern of lobes at the scale quantifed by the

threshold is obtained. Therefore, a scale-dependent study on the lobe hierarchy becomes

feasible (as in Section 3.3). The significance of this method is that the interpretation

process in the experiment can be quantified with an automatic algorithm for the first

time.

62

Figure 3.4: A synthetic demonstration of the temporal constraint: a) based on rules in the interpretation,

only agglomerations on temporarily successive lobes are legitimate in clustering; b) the sequential

cumulated distance from Lobe t1 to Lobe t3 is the distance following the sequential route from t1 to t2 to

t3, promising that d13 is always greater than d12 and d23.

Figure 3.5: The hierarchical clustering result is visualized by a dendrogram. In the synthetic

demonstration, distance 𝑑12 < 𝑑23 < 𝑑13, the dendrogram visualizes a lower scale group (𝑡1, 𝑡2) and a

higher scale group ((𝑡1, 𝑡2), 𝑡3).

63

Figure 3.6: The dendrogram of a complex dataset. Visualization of large datasets with dendrograms is

inconvenient.

Figure 3.7: The reachability plot as a dendrogram alternative to visualize a large dataset. The height of

the bars represent the distance from one observation to all the left neighbors in the plot. The first bar is

set to be the maximum height within this clustering.

64

Figure 3.8: The threshold intersects with bars greater than the chosen scale of interpretation within a

reachability plot. Lobes between two intersections form a lobe at the chosen scale.

Example

An exemplary sample of 36 successive lobes (Fig. 3.9) is selected from a

geomorphic experiment for the demonstration of the algorithm in Stage One. The

dendrogram (Fig. 3.10 a) and reachability plot (Fig. 3.10 b) of the sample are generated

with the weighted mean distance. Within the dendrogram, the 36 lobes are organized by

the order of clustering in the hierarchical clustering and listed along the horizontal axis,

which is the same as the order in the reachability plot. The interesting observation is

that the dendrogram and reachability plot reveal a relatively dense group to the left of

Lobe 5, indicated by a set of lower branch junctions in the dendrogram and a valley in

the reachability plot. The lobe density gradually decreases from Lobe 5 to Lobe 21,

indicating several groups with a fewer number of lobes. Finally, Lobes 1, 2, 3, 4 are

separated from other lobes with a much longer distance in the interpretation space. With

a selected threshold, the sample lobe set is clustered into six groups (Fig. 3.11). The

dense group from Lobe 35 to Lobe 22 are clustered into Group 1, the gradually sparse

lobes are clustered into Group 2 to 4 with a fewer number of lobes, and the sparsest

65

lobes are clustered into Group 5, with a single lobe, and Group 6, with three lobes.

Higher scale lobes formed by each group are also demonstrated in Fig. 3.11. Lobes in

the same group share common MPP and orientation, as well as polygonal overlap and

shape, proving the robustness of the algorithm.

Figure 3.9: A sample lobe sequence of 36 lobes. Color represents lobe location within the sequence,

ranging from 1 to 36.

66

Figure 3.10: a) The dendrogram of the sample lobes, the 36 sample lobes are organized by the order of

being clustered in the clustering algorithm, and listed along the horizontal axis; b) the reachability plot of

the sample lobe; lobes are listed with the same order as in the dendrogram.

67

Figure 3.11: Clusters on the exemplary sample of lobes with a selected threshold. Lobes in each group

form a lobe at the scale of interpretation.

3.2.3 Stage Two – Testing Pattern External Similarity

In this section, a statistical manner of comparing two lobate stacking patterns is

introduced. Lobes in a stacking pattern are characterized by a point pattern of MPPs,

which is further characterized by its probability distribution function of the nearest

neighbor’s proximity for the purpose of reflecting the intensity of the local clustering of

lobes. Probability distribution functions of the nearest neighbor’s proximity are

compared with a hypothesis test. Since normal hypothesis testing techniques are not

robust in cases of small numbers of lobes, a bootstrap two-sample hypothesis test based

on L1-norm measures was devised. The output of the hypothesis test was a p-value

representing the similarity of the patterns. A higher p-value implied the experiment

should be chosen. This method is demonstrated by an example as well.

Characterization of Stacking Patterns

An interesting feature of the parameterization for stacking patterns is that each

parameter has its own distinctive ability to distinguish lobe stacking patterns from

various depositional systems. The most important parameter determining a lobe location

68

is the MPP, because this parameter implies the genetic origin of a lobe. In this sense, a

lobe stacking pattern is converted to a spatial point process of MPPs. The other

parameters, including lobe orientation 𝜃, polygonal distance 𝑝, and lobe shape s are

treated as properties of the MPPs. If a lobe 𝑔 in Stage One is represented as

𝑔(𝑀𝑃𝑃, 𝜃, 𝑝, 𝑠) , in which the four parameters are equivalent features, lobe 𝑔 is

represented as 𝑀𝑃𝑃𝑔(𝜃, 𝑝, 𝑠) in Stage Two, implying orientation, polygonal distance,

and shape are attributes belonging to MMP. Fig. 3.12 a demonstrates a sample set of 60

lobes marked by the MPPs, and Fig. 3.12 b is the spatial point process.

The feature of a lobate stacking pattern that is important to reservoir performance

is the intensity of the local clustering of lobes. The intensity of clustering is detected by

studying the nearest neighboring lobe dissimilarity. Several techniques are available to

characterize the intensity of clustering in a spatial point pattern, such as the empty space

distance function 𝐹, the nearest neighbor distance function 𝐺, and pairwise distance

function 𝐾 (Diggle 2003). The empty space distance is calculated from empty locations

within the domain of the point process to their nearest point. The 𝐹 function is the

empirical cumulative distribution function (ECDF) of empty space distances. The

disadvantage of the 𝐹 function is that the treatment is sensitive to the domain boundary

of the point process. However, the domain boundary for a lobe stacking pattern is the

depositional basin boundary, which is an interpretation. A slight difference in the

interpretation of the basin boundary will affect the resulting 𝐹 function. The 𝐾 function

calculates pairwise distances of points and plots the average number of point versus the

distance of measurement. However, the 𝐾 function is not a cumulative distribution

function, which is inconvenient as an input for the hypothesis test that will be presented

later in this section. The 𝐺 function is the ECDF of the nearest neighbor’s distance in a

point process, which does not depend on boundary measurements and is straightforward

as an input for a hypothesis test. In this work, the concept of the 𝐺 function is applied

to characterize a point process converted from a lobe stacking pattern. For a point

process of a lobe stacking pattern, four 𝐺 functions, 𝐺𝑀𝑃𝑃 , 𝐺𝜃 , 𝐺𝑝, 𝐺𝑠, are calculated

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corresponding to the four parameters denoted earlier. Since other parameters are treated

as attributes of the MPP, the functions are calculated as follows: First, a nearest neighbor

for each lobe is identified in the distance of the MPP, so that 𝐺𝑀𝑃𝑃 is obtained as the

empirical distribution function of the nearest neighbor’s MPP distance. Between each

lobe and its nearest MPP neighbor, proximities for the remaining three parameters are

calculated to obtain the ECDFs, 𝐺𝜃 , 𝐺𝑝 , 𝐺𝑠 . In this manner, a stacking pattern is

characterized by four ECDFs, defined as the statistical characterization of the lobe

stacking pattern, and the similarity between the statistical characterizations of two

stacking patterns is defined as the statistical external similarity. A demonstration of

statistical characterization of the sample dataset (Fig. 3.12 a) is shown in Figure 3.13.

All the stacking patterns must be normalized with procrustes analysis (Section 3.2.2)

for the whole stacking pattern, such that patterns measured at different scales are

comparable.

Figure 3.12: a) An exemplary set of 60 lobes, which is a super set of the sample lobe set in Fig. 3.9. b)

The point process of MPP of lobes in Fig. 3.12 a.

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Figure 3.13: 𝐺 functions (normalized using procrustes analysis) of the spatial pattern in Fig. 3.12.

Bootstrap Hypothesis Test for Two Samples

Since each stacking pattern is characterized by four ECDFs, the problem of

comparing the statistical external similarity between two stacking patterns is now

converted into a two-sample hypothesis test. This test estimates the probability of the

ECDFs from two stacking patterns sharing the same underlying population distribution.

This hypothesis test is performed for each respective parameter. In cases of lobate

depositional environments, the population distributions are considered unknown due to

the complexity of the environment and the lack of data. Hence, the nonparametric

hypothesis testing techniques are appropriate for this work. Several techniques of

nonparametric hypothesis testing are available in the literature, including the

Kolmogorov-Smirnov (K-S) test, the Cramer-Von Mises (Cramer) test, and the

Pearson’s Chi-Square (Pearson’s) test (Corder 2009). The K-S and Cramer tests work

for medium and large datasets, and Pearson’s test is designed for categorical parameters.

In the case of lobate environments, a stacking pattern may be a small-size sample of

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lobes. Thus, a bootstrap (Efron and Tibshirani 1994) two-sample hypothesis test is

devised based on L1-norm measures between ECDFs, such that the test does not rely

on parametric distributions and is robust for small samples.

The test involves a sample 𝐴 of size 𝑀 and a sample 𝐵of size 𝑁 . The null

hypothesis and the alternative hypothesis are

𝐻0: Sample 𝐴 and 𝐵 are drawn from the same cumulative distribution

𝐻𝑎: Sample 𝐴 and 𝐵 are drawn from different cumulative distributions

The testing procedures are as follows

Step 1: Merge the two samples into a pool of size (𝑀 + 𝑁).

Step 2: Calculate the L1 norm difference between Sample 𝐴 and 𝐵 , 𝑑𝐴,𝐵 =

𝑓𝐿1(𝐹�̂�, 𝐹�̂�)

Step 3: Draw a bootstrap sample of size (𝑀 + 𝑁) observations with replacement

from the merged pool.

Step 4: Estimate the ECDF 𝐹�̂� of the first 𝑀 observations and the ECDF 𝐹�̂� of

the remaining 𝑁 observations, and calculate the L1 norm difference 𝑑𝑀,𝑁 =

𝑓𝐿1(𝐹�̂�, 𝐹�̂�).

Step 5: Repeat Step 3 and Step 4 for 𝐾 (e.g., 3000) times and obtain 𝐾 L1-norm

difference values.

Step 6: The p-value of the similarity of accepting the null hypothesis is

𝑝𝑠 = 1 −𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 (𝑑𝑀,𝑁<𝑑𝐴,𝐵)

𝐾 ( 3.7 )

Distinguished from conventional logic of hypothesis testing, this method also

does not require a significance level rejecting the null hypothesis. In this method, the

output p-value is interpreted as the probability that one stacking pattern can be used as

a correct reference to the other. For example, if a high p-value is obtained from the

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hypothesis test with one sample from a geomorphic experiment and one sample from a

real scale depositional system, it is safer to apply inferences from the experiment to

model the real system than using inferences from an alternative experiment with lower

p-values. Since more comprehensive data of initial conditions, intermediate states, and

the final stratigraphy are available in geomorphic experiments, making more precise

quantitative depositional rules or even quantitative geological dynamic models becomes

feasible. To the extent of this study, p-values are used as relative values. A stacking

pattern with higher p-values is interpreted as better than a stacking pattern with lower

p-values. The work does not attempt to interpret the physical meaning of p-values (e.g.,

the physical meaning of 0.4, 0.8 etc.).

Example

Three sets of lobe samples were selected to demonstrate our hypothesis-testing

algorithm (Fig. 3.14). The sizes of Sample 1, 2, and 3 were 13, 16 and 13, respectively.

The sample sizes were designed to be small so that performances of the algorithm on

small samples could be demonstrated. Sample 1 and Sample 2 were from the same

geomorphic experiment and Sample 3 was extracted from a different experiment. Lobes

in Sample 1 and 2 exhibited a high intensity of spatial-temporal clustering, where MPPs

were located in a small region and the lobe orientation varied within a relatively narrow

range. The shape of lobes was generally similar as well. Since Sample 3 was from an

alternative experiment, the stacking pattern was significantly distinguishable from other

samples in the sense that the MPPs were more randomly distributed, the orientation

varied in a wider range, and the shapes demonstrated obvious distinctions from one to

the other. Differences in stacking patterns led to different statistical characterizations

(Fig. 3.15). A visual comparison reveals that all 𝐺 functions of Sample 1 and 2 are more

similar than those of Sample 3. The statistical similarity between Sample 1 and Sample

2, as well as between Sample 1 and Sample 3, were compared using the two-sample

bootstrap hypothesis. Resulting p-values are listed in Table 3.1, in which p-values

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between Sample 1 and 2 are completely higher than between Sample 1 and Sample 3,

proving that the method is capable of identifying similar stacking patterns.

Figure 3.14: a) Lobe Sample 1 of 13 lobes; b) Sample 2 of 16 lobes; c) Sample 3 of 13 lobes. Sample 1

and Sample 2 are from the same depositional system.

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Figure 3.15: 𝐺 functions for all three samples, the samples are normalized using procrustes analysis so

length measurements from different systems are comparable.

P Values Sample 1 vs. Sample 2 Sample 1 vs. Sample 3

𝑃𝑀𝑃𝑃 0.58 0

𝑃𝜃 0.076 0

𝑃𝑝 0.026 0.0075

𝑃𝑠 0.042 0

Table 3.1: P-values of hypothesis testing between samples. Sample 1 and 2 are detected to be similar,

which Sample 3 is different from 1 and 2.

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3.3 Application

The proposed algorithms were demonstrated to be capable of quantifying a

hierarchy of stacking patterns and estimating the statistical external similarity between

stacking patterns. For the application of a scale-dependent analysis of statistical external

similarity between an experiment and a real depositional system, the general workflow

of this method is demonstrated in Fig. 3.16. Inputs to the workflow included a lobate

stacking pattern of a real-scale depositional system and a stacking pattern interpreted

from overhead photos from a geomorphic experiment. In the first step, hierarchical

agglomerative clustering was applied on the experimental stacking pattern. The

hierarchy of experimental lobes was quantified by a dendrogram and reachability plot.

Then, the reachability plot was thresholded regularly on the y-axis (scales). At each

threshold, the lobes were clustered into several groups. Each group was a higher-scale

lobe of the scale of interpretation determined by the threshold. A lobe stacking pattern

was also obtained at the scale. Each experimental stacking pattern was compared to the

real scale pattern with the bootstrap hypothesis test based on L1-norm distance. At every

threshold, a p-value was estimated to measure the similarity. Since the thresholds varied

from very fine to large scales, the p-values were estimated from fine scales to large

scales as well. Finally, the relationship between p-values and the scales of interpretation

was described by a plot. The expected result was that peaks of p-values might be

observed at certain scales of interpretation. Such plots were generated for all four

parameters. If peaks of all plots were at a common scale of interpretation, the

implication was that experimental lobes interpreted at the scale of the peaks are

equivalent to lobes in the real system.

Two stacking patterns from real-scale lobate systems were identified from

sedimentology literatures (Fig. 3.17 a, b). One stacking pattern was from a basin-floor

fan in Kutai Basin, East offshore Kalimantan, Indonesia (A. Saller et al. 2008). The

basin floor fan included 18 lobes, whose length ranged from 3 to 12 km and width

ranged from 1 to 7 km. The other stacking pattern was extracted from the Amazon fan

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(Jegou et al. 2008), including 14 lobes with lengths ranging from 21 to 83 km and widths

ranging from 7 and 25 km. The two patterns exhibited different characteristics in terms

of stacking patterns. In general, lobes in the Kutai basin exhibited more intense overlaps,

while the lobe orientations switched in a relatively large range from one lobe to the

other. Lobe shapes varied from long and narrow to oval and wide. While in the Amazon

fan, very few lobes overlapped the others. Lobe orientations and shapes were both

within a very narrow range. Two hierarchies of lobe stacking patterns were interpreted

from two geomorphic experiments, which were provided by the Sedimentology team,

San Anthony Falls National Laboratory (Fig. 3.17 c,d), Minnesota. Since the

intermediate elevation data was not measured at fine frequencies, the lobe patterns were

interpreted from a fraction of the intermediate photos, in which clear lobate geometries

were identifiable at intervals of 30 seconds. The first experiment (referred to Exp.A in

the following sections) included a sequence of 424 lobes. The sediment cohesion in Exp.

A was designed to be unrealistically low, thus normally channels were short and lobes

randomly aggradated on previous deposits. The second experiment (referred to Exp. B)

included 184 lobes. Because the sediments were designed to be more cohesive, longer

channels were formed during the interval of two avulsions. Thus new lobes tended to

appear near the channel mouth, such that the pattern exhibited more intense stacking.

The reachability plots (Fig. 3.18) quantify the hierarchies of Exp.A and Exp.B. More

small scale valleys are observed in the chart of Exp. A, implying that the sizes of spatial-

temporal clusters are smaller than in Exp. B.

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Figure 3.16: The workflow of scale-dependent similarity analysis. The result is a p-value versus a scale

plot for each parameter.

Figure 3.17: Lobe patterns used in the application: a) stacking pattern of Exp. A; b) stacking pattern of

Exp. B; c) stacking pattern of Kutai basin; d) stacking pattern of Amazon fan.

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Figure 3.18: Hierarchies of experiments: a) hierarchy of Exp. A; b) hierarchy of Exp. B.

All stacking patterns (experimental and real) were normalized with procrustes

analysis so that patterns at different scales became comparable. Then, 60 thresholds at

regular intervals from the finest lobes were taken on the y-axis of the reachability plots

of Exp. A and Exp. B. In the third step, stacking patterns at every scale of interpretation

were obtained by taking the union of lower-scale lobes in each group. Finally, stacking

patterns at each scale were compared to the respective real-scale stacking patterns. In

this manner, p-values were generated as a function of the scales of interpretation. For

the purpose of interpretation, the length unit was converted with respect to the real-scale

system. Assuming a stacking pattern of 𝑁 lobes, the rescaling factor is the RMSD of the

reference real-scale system. The resulting p-values vs. the scale of interpretation plots

comparing both experiments to the Kutai basin are plotted in Fig. 3.19. Fig. 3.20

contains the Amazon fan plots.

The experimental stacking patterns exhibited significant distinctions in p-values

at various scales compared to both real cases. In both cases, low p-values for the MPP

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and 𝜃 were observed spanning most of the plotted range in Exp. A (Fig. 3.19 a and b,

Fig. 3.20 a and b). While higher p-values were observed for the polygonal distance and

the shape in both cases (Fig. 3.19 c and d, Fig. 3.20 c and d), the peaks spanned a wide

range of scales. Since no common scale of high p-values was obtained for all

parameters, Exp. A was interpreted as having no value for providing applicable

information to the Kutai basin or the Amazon fan. On the contrary, plots of Exp. B

versus the real cases were more informative. Peaks of p-values of all plots located within

a narrow range centered at the scale of 15 km (Fig. 3.19). The range of scales for the

four peaks was from 14 km to 17 km (the shaded region in Fig. 3.19), implying that

lobes interpreted within this range in Exp. B are equivalent to lobes in the Kutai basin.

Although peaks were observed in comparing Exp. B to the Amazon fan as well, scales

of the peaks ranged from 14 km to 62 km, covering most portions of available scales of

interpretation in the plots. Stacking patterns for four peaks in Exp. B compared to the

Kutai basin are shown in Fig. 3.21. The stacking patterns for peak scales for the Kutai

basin were not only visually similar but also contained a similar number of lobes (from

30 to 38 lobes). Stacking patterns of Exp. B compared to the Amazon fan are shown in

Fig. 3.22. The numbers of lobes at the peak scales ranged from 22 lobes to 65, which is

significantly different. Hence, it is safer to use Exp. B to provide information to model

the Kutai basin. Lobate geobodies interpreted in the range of 14 km to 62 km in Exp. B

were equivalent to the scale of lobes in the Kutai basin.

In summary, Exp. A, as physically designed, is unrealistic, thus no information

from this experiment are referable to the two real scale systems. Exp. B is more realistic

to all realistic systems than Exp. A in the physical design, thus it has higher similarity

than Exp. A to both real scale cases. However, Exp. B cannot provide referable

information for the Amazon fan either. Because the petrofacies information of neither

Kutai basin nor Amazon fan were discussion in the publications, we can only interpret

this in the perspective of lobe spatial distribution. Generally, channels in the Amazon

fan are obviously longer than those in the Kutai basin and in Exp. B and lobe distribution

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are sparse relative to lobe distributions in the Kutai basin and Exp. B, hence, the high

p-values are inconsistent in all four parameters.

Figure 3.19: P-values versus scales of interpretation plots with respect to the Kutai basin. The distance

measurements are rescaled with RMSD of the Kutai basin. Peaks of p-values are marked out by the black

dots and a narrow range for scales of interpretation with the highest p-values for all four parameters

(shaded regions) is identified. a) MPP; b) Orientation; c) Polygonal distance; d) Shape.

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Figure 3.20: P-values versus scales of interpretation plots with respect to the Amazon fan. The distance

measurements are rescaled with RMSD of the Amazon fan. Peaks of p-values are marked out by the black

dots, but the peaks vary in a wide range of scales. a) MPP; b) Orientation; c) Polygonal distance; d) Shape.

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Figure 3.21: Stacking patterns from Exp. B at peaks of p-values with respect to the Kutai basin: a) patterns

at the scale of interpretation with the highest p-value for the MPP (38 lobes) at the scale of 13.8km; b)

patterns at the scale of interpretation with the highest p-value for 𝜃 (35 lobes) at the scale of 15.1km for;

c) patterns at the scale of interpretation with the highest p-value for polygonal distance (30 lobes) at the

scale of 15.8km; d) patterns at the scale of interpretation with the highest p-value for shape (28 lobes) at

the scale of 16.8km.

Figure 3.22: Stacking patterns from Exp. B at peaks of p-values with respect to the Amazon basin: a)

patterns at the scale of interpretation with the highest p-value for the MPP (22 lobes) at the scale of 62km;

b) patterns at the scale of interpretation with the highest p-value for 𝜃 (26 lobes) at the scale of 47km for;

c) patterns at the scale of interpretation with the highest p-value for polygonal distance (30 lobes) at the

scale of 42km; d) patterns at the scale of interpretation with the highest p-value for shape (65 lobes) at

the scale of 14km.

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3.4 Chapter Summary

This chapter presented a workflow of identifying a proper geomorphic experiment

and the proper scale to interpret the experiment with respect to a stacking pattern

interpreted from a real-scale depositional system. The workflow is based on data mining

and statistical analysis, and is proved to be capable of generating reasonable results by

applications on two experiments and two real systems. In addition, several techniques

to quantify geological information, such as the stacking pattern and hierarchy of lobes,

were introduced in the workflow, which are useful tools for quantitative studies on

lobate systems.

The proposed workflow also leads to future work. First, the stacking patterns were

quantified by four parameters, but not all of them could effectively distinguish one

stacking pattern from another. One potential improvement could be made on the

measurement of polygonal proximity. In this study, the p-values versus the scale of

interpretation plots of polygonal proximity, measured using the Hausdorff distance (Fig.

3.19 c and Fig. 3.20 c), were noisy and not as informative as the other three parameters.

A possible reason is that the areal proximity measurement should be capable of catching

the degree of overlap of two small lobes, which is not considered by Hausdorff distance.

Second, the hierarchy of lobe systems was quantified by a dendrogram or reachability

plot, which is considered as a quantitative summary of the system. It would be worth

developing a technique for measuring the differences between two dendrograms, which

would reflect the overall statistical similarity between two lobate systems. Third, the

current L1-norm measurement based bootstrap hypothesis test was designed for samples

of a small size. The method outperformed a K-S test with small samples; however, p-

values estimated with this bootstrap test might be underestimated compared to those

estimated with the K-S test. Moreover, the robustness of this hypothesis testing method

should be further estimated through applying it to stacking patterns from a large number

of experiments, such that the situations in which the p-values are reliable can be

determined. Finally, the model identified a scale of interpretation within an experiment

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equivalent to lobes at the seismic scale of a real system. Since the interest of static

reservoir modeling is to estimate the subseismic scale lobe stacking patterns, available

in the quantified experimental hierarchy, the next step would be to develop a simulation

algorithm that honors the hierarchical information.

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Chapter 4

HIERARCHICAL SIMILARITY OF

LOBES: SIMULATION AND

COMPARISON

4.1 Introduction

The results of Chapter 3 are a set of lobes from one experiment that is the most

similar to a specific real depositional system within multiple optional experiments. A

reasonable assumption is that the geological dynamic process that produces this

similarity is itself also similar to the process that generates the real lobe stacking pattern.

Based on this assumption, this chapter is dedicated to an alternative implementation of

lobe migration dynamic processes, designed to honor information provided by the

chosen experimental lobe stacking pattern. The implementation was designed to involve

the least number of empirical coefficients so that the model is favorable for uncertainty

estimations. As indicated earlier, the frequency of three-dimensional intermediate

topography measurement is not sufficient in this experiment, thus the method developed

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in this chapter only considers lobe stacking patterns extracted in the two-dimensional

horizontal plain.

The most valuable information from an experimental lobe stacking pattern is the

sequential migration behavior of lobes. Current lobe migration rules (Fig. 2.16) in

surface-based models determine placement of lobes based on conceptual understandings

about depositional systems, normally interpreted from field and seismic surveys and the

general physical rules. The lack of intermediate topography and the limited amount of

lobes maintained in field-scale depositional systems limit more precise understanding

of the depositional process and thus more precise quantitative depositional rules.

Current surface-based models with conceptual depositional rules are sufficient for

applications in the verification or demonstration of hypothesized conceptual rules.

However, in applications for engineering purposes, e.g. conditioning, uncertainty

estimation, history matching, disadvantages of conceptual depositional rules are

obvious. First, the computational implementation of conceptual rules involves a

modeler’s subjective decisions regarding conversion functions from qualitative

concepts, which are normally sensitive to model realizations. Therefore the choice of

conversion functions directly increases the uncertainty estimations. On one hand, the

ranges of values of empirical coefficients of the conversion functions cannot be

narrowed down because there is no sufficient data to quantitatively calibrate the

algorithms of conceptual rules. To test uncertainty of a wide ranges of values is an

incredible workload. For example, the probability component reflecting impacts of the

basin source (Fig. 2.16) is converted from a distance-to-basin-source map, in which the

value in every cell of the modeling grid is its distance to the basin source point. The

conversion function from the distance to the probability is the modeler’s decision, such

as logarithmic, gaussian, or exponential functions of the inversed distance etc. Each

conversion function has its own coefficients directly affecting the simulated spatial

distribution of lobes, e.g. the mean and variance for the gaussian function. Considering

there are usually multiple components in the conceptual depositional model, a handful

of scenarios and tens of empirical coefficients must be considered in the uncertainty

87

estimation. On the other hand, the interactions between parameters are extremely

unpredictable in a computational implementation of conceptual models with too many

empirical parameters. Thus, the response of a surface-based model is probably

dominated by noise, leading to difficulty in any inversion problem with the model, such

as conditioning. Finally, conceptual depositional rules in current surface-based model

implementations rarely attempt to account for any specific input patterns, such as the

stepwise characteristics or overall hierarchy of a lobe migrating sequence from an

experiment. The objective of this chapter is to present an improved computational

implementation for lobe migration in order to minimize the subjective choices of

quantification functions and empirical coefficients with a simple algorithm that is

capable of using prior information from an input lobe sequence.

The precise lobe migration sequence provides more specific information on the

depositional process, which is valuable in the design and calibration of a lobe migration

algorithm, such that the algorithm design can be improved, e.g. to reduce the number of

parameters, to test and fix proper values of empirical coefficients. The essential

meaningful information provided by a lobe migration sequence is the stochastic

relationship between intermediate topography and lobe migration. Since the migration

sequential information is limited in the horizontal plain, the first task is to define a

characterization of the two-dimensional lobe migration sequence such that the surface-

based model can account for geologically meaningful information from the input

sequence and the intermediate topography (surfaces) in every step of the simulation. In

this chapter, the lobe migration is described with a correlated random walk (CRW)

(Section 4.2), in which the stepwise behaviors and overall hierarchies of the input

stacking patterns are statistically reproduced. In the CRW-based strategy, empirical

coefficients are used as infrequently as possible. The new strategy, demonstrated here,

is an implicit control on the hierarchy of the lobe MPP. A dissimilarity measure is

proposed to validate the reproduction of the MPP hierarchy between two lobe stacking

patterns, with which it is proved that MPP hierarchies of model realizations are

effectively constrained (Section 4.3). The significance of the CRW-based lobe

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migration strategy is that the prior depositional dynamic process model can be updated

effectively by replacing input migration sequences analogous to updating training

images (conceptual models of finalized petrofacies) in multiple-point statistics, which

is meaningful in the rapid updating strategy for history matching (Caers 2013).

4.2 A Lobe Migration Model with Input Lobe Sequence

This section illustrates the implementation of a CRW-based lobe migration

mechanism. As aforementioned, the model is expected to honor an input lobe stacking

pattern. Since the input experimental stacking pattern is normally at a different spatial

scale to the simulated depositional basin, a normalization and rescaling factor is required

to convert any spatial measurements from input experiments to the real system to be

modeled. The Root Mean Square Distance (RMSD), a statistical measure of the scale

of a spatial pattern, is proposed to be the normalization factor (Section 4.2.1). Instead

of using the pairwise interlobe proximity of the finalized stacking patterns (Chapter 3),

statistics of stepwise pairs in the input lobe migration is considered in this section (Fig.

4.1).

Temporal and spatial information in the horizontal two-dimensional domain of

the sequence was precisely recorded in the inputs; however, the intermediate

topographic information was available at every step of the simulation besides two-

dimensional spatial and temporal information. This led to the requirement of accounting

for two types of information: the conceptual compensational stacking rules in response

to the intermediate topography and the precise statistics of migration in the two-

dimensional horizontal domain. In the method, a bifactor modeling scheme was

proposed to combine both types of information (Section 4.2.2). While characterizing

and simulating the lobe migration pattern, the focus was on the MPP of the lobes and

the lobe orientation (Section 4.2.3 – 4.2.4), which were the most important parameters

of the spatial distribution of the lobes. Simulation of the polygonal distances and shape

similarities were not considered in this work. Finally, an exemplary run of the model

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was performed to demonstrate the new surface-based model with new lobe migration

mechanism (Section 4.2.5).

Figure 4.1: a) The plane view of an experimental stacking pattern with the high p-values in Chapter 3,

including 38 lobes. b) The sequence view of the experimental stacking pattern, which is explicitly input

to the algorithm in this chapter.

4.2.1 Root Mean Square Distances: The Scaling Factor for Stacking Patterns

The input stacking patterns were obtained from geomorphic experiments, in

which the length scale was much smaller than that of a real-scale depositional system.

As an extension of the strategy of measuring statistics of dimensionless ratios presented

in Chapter 1, using a uniform scaling factor that normalized length measurements in

both the input stacking pattern and the simulation realizations for a dimensionless space

was proposed. The lobe migration sequence was modeled and simulated in the

dimensionless space and was rescaled to the unit of the real depositional system after

the simulation. The proposed uniform scaling factor was the RMSD. RMSD is a popular

statistical measure of the scale of a set of data and is adopted in procrustes analysis for

the analysis of shapes of different sizes (Dryden and Mardia 1998; Kendall 1989). For

this application, the Mean Square Distance (MSD) was calculated between any point

𝑝𝑖(𝑥𝑖 , 𝑦𝑖) on the interpreted basin boundary to 𝑝𝑐(�̅�, �̅�), the geometric centroid of the

basin. The RMSD was simply the square root of the MSD, such that the value was

converted from a real unit to a length unit (Eq. (4.1)). Any distance measurement, 𝑑𝑒𝑥𝑝,

in the experiment was converted to the dimensionless space by scaling it with the RMSD

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of the input stacking pattern 𝑅𝑀𝑆𝐷𝑒𝑥𝑝, the simulated realizations in the dimensionless

space were converted to the real system scale with the RMSD of the real basin

𝑅𝑀𝑆𝐷𝑟𝑒𝑎𝑙 (Eq. (4.2) – (4.3)).

𝑀𝑆𝐷 = √∑ [(𝑥𝑖−�̅�)2+(𝑦𝑖−�̅�)2]𝑖

𝑘 ( 4.1 )

�̂� = 𝑑𝑒𝑥𝑝

𝑅𝑀𝑆𝐷𝑒𝑥𝑝 ( 4.2 )

𝑑𝑟𝑒𝑎𝑙 =�̂�

𝑅𝑀𝑆𝐷𝑟𝑒𝑎𝑙 ( 4.3 )

Figure 4.2: Root Mean Square Distance (RMSD) as the scaling factor for stacking patterns. RMSD is

determined by the averaged squared distance from the geometric centroid to the interpreted basin

boundary. It is a measure of the scale of a stacking pattern and is used as the scaling vector to normalize

stacking patterns. All length measurements and coordinates from the experiment are normalized by the

RMSD of the experimental pattern, and the simulation is performed in the dimensionless space. The

RMSD of the field basin can be used to scale the realizations to the basin scale. Refer to Eq. (4.1) – (4.3).

4.2.2 The Bifactor Modeling Scheme

The first task of modeling a lobe migration sequence was the strategy of

accounting for two types of information, including precise statistics from the input

horizontal spatial lobe migration sequence and the conceptual compensational stacking

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rule in response to the intermediate topography at every step of simulation. From the

perspective of the depositional process, the spatial distribution of sediments was

governed by the compensational stacking rule, which was a stochastic process

endeavoring to transport sediments into lower regions such that the sediment basin was

turned into a horizontal plain. On the contrary, accounting for an input lobe sequence

implied modeling the quantitative relationship between intermediate topographies and

the spatial location of a new lobe. Since the intermediate topography information is not

available in the experiment data used for this chapter, building such a stochastic

relationship is impossible; hence, the lobe migration mechanism in this chapter only

describes the movement of the input lobe sequence without considering the real

relationship between intermediate topographies and lobe locations. This scheme is in

analogy to the kinematics in classical mechanics, in which only the motions, but not the

cause of motions, are considered. The effect of the intermediate topography was

considered by combining the conceptual compensational stacking rule at every step of

simulation. In detail, the component of conceptual compensational stacking rule always

tend to place a new lobe toward the steepest direction away from the previous one, while

the input lobe sequence component determine the next MPP location based on statistical

behavioral characteristics of the input. The combined modeling scheme of the two

factors is formulated in Eq. (4.4). The combination weight was the first empirical

coefficient introduced in this modeling strategy to reflect the modeler’s judgement on

the importance of the two factors.

𝑔 = 𝑏 ∗ 𝑔𝑝𝑎𝑡𝑡𝑒𝑟𝑛 ⊕ (1 − 𝑏) ∗ 𝑔𝑐𝑜𝑚𝑝 ( 4.4 )

where 𝑔𝑝𝑎𝑡𝑡𝑒𝑟𝑛 is the prior information from the input lobe sequence; 𝑔𝑐𝑜𝑚𝑝 is the prior

information from the conceptual compensational stacking rules; ⊕ represents the

combination of two types of priors; 𝑏 is a combination weight, 𝑏 = 1 implies total

control by the lobe sequence, 𝑏 = 0 implies total control by the compensational

stacking rules.

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4.2.3 The Input Stacking Pattern Factor

Parameterization of Lobe Sequences

The basic requirement of the characterization strategy was the simplicity,

implying both the simplicity of parameterization of the input movements and the

resulted simplicity in the mechanism implementation. A geological model has to be as

simple as possible to reduce the workload in uncertainty estimations and history

matching. Another requirement is that the characterization of a lobe migration sequence

should produce geologically reasonable features, such that the simulation is not over

constrained to the input sequence. In lobate environments, the depositional process is

featured by the stepwise relative movements of lobes, such as progradation,

retrogradation, and lateral shift. Progradation and retrogradation are the successive

basinward or landward trends of lobe migration movements, in which a new lobe

location is defined relative to the previous one. The basic assumption behind studying

relative lobe movements is the directional persistent behavior. A conceptual explanation

for the directional persistence assumption, also for the previous lobe location

component in Chapter 2 (Fig. 2.16), is that a sediment flow tends to follow the channel

through which the previous flow went. When the new flow passes through the channel-

to-lobe transition point of its previous lobe, the local slope, flow condition, and strength

of the channel levee determine either the formation of a new lobe or the continuation of

channelization toward the downstream direction. Generally speaking, the change of

slope around the channel mouth is drastic, and the channel levee is not well formed.

Hence, a new lobe normally occurs in the intermediate vicinity of the previous lobe.

Some of the features, such as the migration direction (progradation or retrogradation)

and lateral shifting direction, are related to the intermediate regional topographic and

fluid conditions. However, other features, such as the statistics of the stepwise migration

length and lateral shifting angle of the lobe orientation, can distinguish one lobate

system from another, although the physical explanation remains unclear. For example,

some lobate systems are characterized by sustained progradation or retrogradation with

frequent short migration lengths and a small shifting angle, while others are

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characterized by an almost completely random lobe distribution with frequently large

migration steps and shifting angles. A strategy to characterize lobe migration behavior

with the direction persistence assumption and orientation changes of every single lobe

is presented here. The strategy also assumes that lobe migration behavior is stationary

and homogeneous within an input lobe sequence. Consistent to the workflow in Chapter

3, MPP is used as the primary parameter indicating the location of a lobe. Thus, the

migration distance ∆𝑟 from the current MPP to the next is measured as the stepwise lobe

migration distance (Fig. 4.3 a). A MPP migration shifting angle, ∆𝜃𝑀𝑃𝑃, is required to

determine the coordinate of the new MPP. ∆𝜃𝑀𝑃𝑃 for the step 𝑡 + 1 is the angle between

the previous MPP migration direction, defined as the vector from 𝑀𝑃𝑃𝑡−1 to 𝑀𝑃𝑃𝑡, and

the new MPP migration direction, defined as the vector from 𝑀𝑃𝑃𝑡 to 𝑀𝑃𝑃𝑡+1 (Fig. 4.3

b). ∆𝜃𝑀𝑃𝑃 is measured from 0 to 2 ∗ π for the purpose of recording the precise

movement pattern. Besides MPP migration, the lobe orientation is also an important

parameter determining the exact location of a lobate geobody. Lobe orientation is

defined as the unit vector from MPP to the geometric centroid of a lobe in the horizontal

two-dimensional plain. Conceptually, the relative movement between lobe 𝑡 and lobe

𝑡 + 1 is affected by the basinwise topography and overall flow orientation, but in the

meantime it is stochastic in response to the regional topography. Instead of measuring

it relative to the previous lobe orientation, the lobe orientation shifting angle is measured

relative to the interpreted overall basin flow orientation (Fig. 4.3 c), defined by the unit

vector from the basin source point to the two-dimensional geometric centroid of the

basin. The lobe orientation shifting angle, ∆𝜃𝐿, is measured from 0 to π. For each input

lobe migration sequence, ∆𝑟 , ∆𝜃𝑀𝑃𝑃 , and ∆𝜃𝐿 are measured through the sequence,

resulting in three ECDFs.

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Figure 4.3: MPP migration and lobe relative (to the basin flow orientation) shifting angle are modeled in

this chapter. Scale versus stacking pattern plots of the area distance and the shape are noisy, thus the use

of them as inputs requires further studies. a) ∆𝑟 , the stepwise migrating distance of MPP; b) ∆𝜃𝑀𝑃𝑃, the

stepwise MPP migrating orientation, measured from 0 to 2π; c) ∆𝜃𝐿, the stepwise lobe relative (to the

basin flow orientation) orientation, measured from 0 to π.

Correlated Random Walk (CRW) for MPP Migration

The lobe sequence characterization with CDFs of the MPP migrating distance ∆𝑟,

and migration orientation shifting angle ∆𝜃𝑀𝑃𝑃, naturally leads to the application of the

CRW as the modeling technique. CRW broadly refers to the homogeneous and

stationary sequential random point movement, in which the location of a new point is

determined by the stepwise migrating distance and the turning angles from a previous

location and moving direction (Fig. 4.4). In the most common form of CRW, the

stepwise migration distance follows a non-negative right-skewed parameterized

probability distribution, i.e. weibull distribution (Eq. (4.5)), and the turning angle from

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a previous moving direction follows a symmetric distribution, in which values are

around zero, i.e., wrapped cauchy distribution (Eq. (4.6)).

𝑓(𝑟) = 𝑎(𝑟)𝑎−1𝑒−(𝑟)𝑎 ( 4.5 )

where the shape parameter 𝑎 controls the shape of the distribution;

𝑓(𝜃) = 1

2𝜋(

1−𝑘2

1+𝑘2−2𝑘𝑐𝑜𝑠[𝜃−𝐸(𝜃)]) ( 4.6 )

where 𝑘 ∈ [0,1] controls the range of the turning angle, 𝐸(𝜃) is the mean direction;

Figure 4.4: The idea of Correlated Random Walk (CRW). CRW simulates spatial point sequences. The

point at time 𝑡 + 1 is determined by deviations from the point at 𝑡 by a moving distance ∆𝑟 and an angle

∆𝜃𝑀𝑃𝑃 deviating from the previous MPP moving orientation. ∆𝑟 and ∆𝜃𝑀𝑃𝑃 are modeled as stochastic

variables with cumulative distribution functions (CDFs). The choice of CDFs depends on the problem.

CRW is applied to model MPP migration in this work and controlled by empirical CDFs from the input

lobe sequence.

An increment of 𝑎 in the weibull distribution, and 𝑘 in the wrapped cauchy

distribution, increases the probability of getting larger ∆𝑟 and ∆𝜃𝑀𝑃𝑃 and thus

introduces more randomness in the simulated points (Fig. 4.5). Since CRW is simple

and has been widely studied, the technique has been applied in various scientific

problems. The first CRW model was proposed to analyze and model the migration

behavior of butterflies (Kareiva and Shigesada 1983). Since then, and along with the

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application of new data obtaining techniques, such as remote sensing image, GPS, etc.,

the CRW model has been widely applied or modified based on the Kareiv-Shigesada

model in studies of animal migration behaviors (Bergman, Schaefer, and Luttich 2000;

Firle et al. 1998; Forester et al. 2007; Fortin, Morales, and Boyce 2005; Fortin et al.

2005; Fortin et al. 2005; Marell 2002). In a relatively sophisticated example (Fortin,

Morales, and Boyce 2005), a CRW with bias to a topography model was applied to

study the migration movement of elk. The assumption of their model was that average

direction 𝐸(𝜃) of the probability distribution of the next turning angle of the elk

wandering randomly was a function of two factors: the directional persistence factor

and the steepest slope direction. Therefore, they modeled the average moving direction

𝐸(𝜃) of the wrapped cauchy distribution for sampling the next turning angle as a

weighted mean of the steepest slope direction and the previous moving direction. The

distribution functions were fitted to data from 21 elk-migration routes. The idea of CRW

with bias to a topography model effectively incorporated the impacts of topography on

the migration movement with a simple technical framework, which was adjusted and

extended here for the purpose of accounting for topography in the lobe migration

mechanism model. The primary modification is that the ECDFs of the MPP migration

distance replace the weibull distribution. To determine the MPP migration orientation-

shifting angle, the mean migration direction was determined by weighted averaging of

a sampled value from the input ECDF of ∆𝜃𝑀𝑃𝑃 and the steepest slope direction. The

weighted mean moving direction was used as an average direction in the wrapped

cauchy distribution for sampling the final lobe migration orientation. The reason for

maintaining the wrapped cauchy distribution in the model was to keep sufficient

randomness in the model. The second empirical coefficient 𝑘 is introduced in the

wrapped cauchy distribution to control the variability of the distribution, or the strength

of directional persistence.

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Figure 4.5: The randomness in CRW. The migration distance ∆𝑟 is usually modeled by a weibull

distribution and ∆𝜃𝑀𝑃𝑃 is modeled by a wrapped cauchy distribution. Both distributions provide

parameters to control the means and shapes. Distributions with higher probability to large ∆𝑟 and ∆𝜃𝑀𝑃𝑃

lead to more randomly distributed point sequences, while higher probability for small ∆𝑟 and ∆𝜃𝑀𝑃𝑃 lead

to sequences consistently migrating in the same direction.

4.2.4 The Modeling Algorithms

The bifactor lobe migration mechanism includes two parts: the MPP model (Fig.

4.6 a) and the lobe orientation model (Fig. 4.6 b), both of which are described in this

section.

Algorithm 4.1 𝐌𝐏𝐏 Movement

This algorithm includes two steps. In the first step, the steepest slope direction is

calculated at the sampled MPP migration distance from the previous MPP. In the second

step, the migrating direction reflecting the input lobe sequence is calculated with CRW.

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The two orientations are averaged with a combination weight, which represents the

modeler’s decision.

Step 1: Sample ∆𝑟 from the lobe migration distance ECDF of the input lobe

migration sequence;

Step 2: Find points 𝑝𝑘,𝑡 such that distances 𝑑𝑘,𝑡 from the current MPP 𝑀𝑃𝑃𝑡 to all

𝑝𝑘,𝑡 satisfying 𝑑𝑘,𝑡 = ∆𝑟 ;

Step 3: Find the point 𝑝𝑠𝑡𝑒𝑒𝑝,𝑡 within 𝑝𝑘,𝑡, such that the elevation change from the

previous MPP to 𝑝𝑠𝑡𝑒𝑒𝑝,𝑡 is the steepest;

Step 4: Find the steepest descending direction 𝜃𝑠𝑡𝑒𝑒𝑝𝑒𝑠𝑡,𝑡 by the unit vector from

𝑀𝑃𝑃𝑡 to 𝑝𝑠𝑡𝑒𝑒𝑝,𝑡;

Step 5: Sample the migration orientation shifting angle ∆𝜃𝑀𝑃𝑃,𝑡 from the ECDF;

Step 6: Calculate the new orientation reflecting the input lobe pattern by 𝜃𝑀𝑃𝑃,𝑡 =

𝜃𝑀𝑃𝑃,𝑡−1 + ∆𝜃𝑀𝑃𝑃,𝑡;

Step 7: Calculate the new mean orientation for the wrapped cauchy distribution

𝐸[𝜃]𝑀𝑃𝑃,𝑡 = 𝑏 ∗ 𝜃𝑀𝑃𝑃,𝑡 + (1 − 𝑏) ∗ 𝜃𝑠𝑡𝑒𝑒𝑝𝑒𝑠𝑡,𝑡;

Step 8: Sample the real migration direction with the wrapped cauchy distribution

and 𝐸[𝜃]𝑀𝑃𝑃,𝑡

Algorithm 4.2: Lobe Orientation

Step 1: Sample the lobe shifting angle ∆𝜃𝐿,𝑡 from the ECDF;

Step 2: Calculate two possible orientations with 𝜃𝐿1,𝑡 = 𝜃𝐵𝑎𝑠𝑖𝑛 + ∆𝜃𝐿,𝑡 and

𝜃𝐿2,𝑡+1 = 𝜃𝐵𝑎𝑠𝑖𝑛 − ∆𝜃𝐿,𝑡+1;

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Step 3: Choose the direction 𝜃𝐿,𝑡 , such that the elevation changes along the

direction are steeper;

Figure 4.6: The implementation of the combined lobe migration mechanism: a) the bifactor model

combining MPP migration information from the input lobe sequence and the conceptual compensational

stacking rule in response to the intermediate topography in the simulation, refers to Algorithm 4.1. b)

determining the lobe orientation, refers to Algorithm 4.2.

4.2.5 Example

A run was performed on the new lobate model with the CRW-based lobe

migration mechanism. Primary parameter values of the simulation are documented in

Table 4.1. The purpose of this simulation was to demonstrate the capability of

generating a realistic lobe migration sequence and more complex stratigraphy than the

model with the conventional conceptual lobe deposition process demonstrated in

Chapter 2. For the purpose of simplification, the lobe size was set to be constant. The

basin source was stochastically sampled within a small segment located in the middle

of an edge, and the overall basin flow orientation was perpendicular to the edge of basin

source at the midpoint (Fig. 4.7). The initial topography was a plain surface. The input

lobe migration sequence is demonstrated in Fig. 4.1, from which the ECDFs of the MPP

migration distance, MPP migration orientation-shifting angle, and lobe orientation-

shifting angle were extracted (Fig. 4.8). The simulation was designed to imitate the

experiment, so no normalization was performed on the input lobe stacking pattern. The

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directional persistence strength control factor for the wrapped cauchy distribution was

set to be 𝑘 = 0.6, implying that the lobe distribution process was moderated between

being completely random and completely directionally persistent. The combination

factor 𝑏 was set to be 0.5, implying that precise statistics from the input lobe sequence

and the conceptual compensational stacking rule accounted for half of the lobe

migration mechanisms. 100 lobes were simulated in every realization. The intermediate

topographies of five steps from one realization are demonstrated in Fig. 4.9. The

depositional process started from the top edge of the model grid, and the first lobe

complex was formed at the top-right portion of the grid from Lobe 1 until Lobe 20. The

process started to laterally shift from Lobe 20 and filled the up-left region of the grid

with a second lobe complex at around Lobe 40. After some random filling around Lobe

50, progradation was observed at Lobe 60 toward the distal empty region of the

modeling grid. In the following steps up to Lobe 100, a third lobe complex was in the

process of formation. The lobe depositional sequence is visualized at three vertical

sections from the proximal to distal in Fig. 4.10. Compared to the lobe sequence section

generated by the conceptual depositional rules demonstrated in Fig. 2.26, more natural

and realistic spatial-temporal clusters of lobe were observed. Finally, the ECDFs of the

statistics from the input lobe migration sequence were compared to that from the

realization (Fig. 4.11). Since the MMP migration orientation was a weighted average of

two factors, this ECDF is not reproduced in the realization. However, reproductions of

the migration distance and lobe-shifting angle were satisfying.

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Simulated Object Lobe only

Modeling Grid 200 cells by 200 cells

Cell Size 1

Lobe Length 𝐿 = 120 (constant)

Basin Source 𝑥 = 0; 94 < 𝑦 < 106 Basin Flow Orientation

Perpendicular to the edge of basin source

Initial Topography Plain

𝑘 (wrapped Cauchy distribution) 𝑘 = 0.6

Combination Factor 𝑏 = 0.5 Number of Lobes 𝑁 = 100

Table 4.1: Values of primary parameters in the simulation.

Figure 4.7: Model grid setting for the test simulation. The basin source sample within a small section

around the middle point of an edge (refer to Table 4.1) and the basin flow orientation were set to be

perpendicular to the edge of the basin source toward the modeling grid.

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Figure 4.8: ECDFs from the inputs lobe migration sequence.

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Figure 4.9: Samples of intermediate topographies of a realization with 100 lobes. Several lobe complexes

following the compensational stacking rule are observed in the simulation.

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Figure 4.10: One realization of 100 lobes and the proximal, medial, and distal depositional sequence plot.

Lateral shifting and spatial-temporal clustering are observed.

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Figure 4.11: The reproduction of input ECDFs of ∆𝑟 , ∆𝜃𝑀𝑃𝑃 , and ∆𝜃𝐿𝑜𝑏𝑒 . Relative MPP migrating

orientation ∆𝜃𝑀𝑃𝑃 is not reproduced due to the combination with in-situ compensation stacking rules.

4.3 Hierarchical Similarity

The ECDFs of the MPP migration distance, MPP migration orientation-shifting

angle, and lobe orientation-shifting angle only characterize the sequential pairwise

information of the migration process. Nevertheless, an input lobe sequence also

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provides overall information on the lobe stacking pattern, such as lobe hierarchies

described by the dendrogram or reachability plots. In Chapter 3, the dendrogram of a

stacking pattern was calculated with the sequential cumulated weighted mean distance

over all four parameters, measuring pairwise lobe proximity. Since the weights are

simply chosen by the modeler, reproducing the dendrogram of a stacking pattern is

essentially the problem of reproducing dendrograms for each of the parameters: MPP

distance, lobe angle, polygon distance, and shape dissimilarity. The optimal lobe

migration mechanism should be capable of constraining all four dendrograms, which is

complicated. The reason is that characterizing dendrograms requires obtaining all the

simulated lobes, while the lobe migration is a forward process in which the final pattern

information is unavailable before a simulation is completed. Thus, accounting for the

final stacking pattern hierarchies is a computationally intensive and inefficient inversion

problem in most situations, which is unfavorable in the geological modeling stage of a

reservoir development project. This section demonstrates that the input sequential

statistics in the CRW-based mechanism are an implicit control of overall lobe

hierarchies in the forward simulation process (Section 4.3.2). Thus, all realizations with

the new lobate model are hierarchically similar to the input lobe migration sequence

(Section 4.3.3), which is a new feature not available in the model with conceptual

depositional rules presented in Chapter 2. The prerequisite of comparing hierarchies is

a dissimilarity measure quantifying the proximity with reasonable geological

interpretations, introduced in Section 4.3.1.

4.3.1 Quantitative Measure of Hierarchical Similarity

For the purpose of studying the hierarchical similarity of the lobe MPP, the plane

and perspective view of the input lobe migration sequence and of one simulated

realization were plotted. The input lobe sequence included 38 lobes (Fig. 4.1), and the

realization involved lobes from Step 1 to Step 40 in the example, demonstrated in

Section 4.2.5. An observable difference between the two sequences was that the shifting

angle of the MPP migration direction in the input sequence (Fig. 4.12) was more

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random, although that in the simulated realization (Fig. 4.13) was more directionally

persistent, caused by the directional persistence controlling parameter 𝑘 and the

combination factor 𝑏. The lobe hierarchical similarity based on the sequential migration

distance was controlled in the model, given the difference in the MPP migration

direction caused by 𝑘 and 𝑏. In this dissertation, hierarchies of MPP migration distances

are quantified by dendrograms (Fig. 4.14) generated by agglomerative hierarchical

clustering with sequential cumulated distances, so the problem of measuring the

hierarchical similarity is converted to defining a dissimilarity measure function to

quantify the proximity between dendrograms of the patterns.

A dendrogram is formally defined as a rooted, terminally-labeled, and weighted

tree in which all terminal nodes are equally distant from the root (Lapointe and Legendre

1995). A dendrogram consists of three components: topology, label, and weights (Fig.

4.15). Topology is the bifurcation structure between leafs only, the label links each leaf

to a lobe in addition to the topology, and the weight considers the height of the branches,

accounting for the distance between clusters. Comparison of denrograms includes two

parts, a descriptor of the dendrogram and a function measuring the difference of the

descriptor. The most popular dendrogram descriptor is the cophenetic distance (Sokal

and Rohlf 1962), defined as the interleaf distance within the plot. In a dendrogram, the

cophenetic distance between two leaves is the height at which two leaves merge into a

branch (Fig. 4.16). The cophenetic distance only considers the label and weights, not

topology. The other popular descriptor is the topological distance (Phipps 1971), which

considers topological properties only, defined as the number of junctions counted along

the route from one leaf to another. Three variations of cophenetic distance and

topological distance are proposed (Podani and Dickinson 1984), all of which are

considered together as a multivariate descriptor. Functions measuring the difference

between descriptors in the literature, unfortunately, are all designed to compare two or

more dendrograms generated from the same datasets of n observations. For example, if

two dendrograms are obtained from the same dataset with different hierarchical

clustering algorithms, two cophenetic distance values can be calculated for every

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observation. Pearson’s correlation coefficient between the two sets of cophenetic

distances of size n is estimated to quantify the similarity between the two dendorgrams.

Known as the cophenetic correlation coefficient (Sokal and Rohlf 1962), the technique

is prevalent and is applicable to other descriptors.

Since the compared dendrograms in this study were generated from lobe

sequences normally composed by a different number of lobes, the difference function

was defined as the ECDF of the cophenetic distance of a dendrogram. There were two

advantages of the new descriptor. First, the ECDF of the cophenetic distance

characterized the intensity of the clustering effect of the stepwise lobe migration; thus,

the comparison was based on the statistical features of geologically interpretable

clusters of a migration sequence, e.g. progration, retrogradation, and aggradation.

Second, the ECDFs did not require an equal number of observations in the two

dendrograms, such that hierarchies of sequences with different numbers of lobes can be

compared. Once the dendrograms were converted to the ECDFs of their cophenetic

distances, the L1-norm difference was used to measure the dissimilarity between two

distribution functions. The ECDFs of the cophenetic distances of both input lobe

sequences and realization lobe sequences are demonstrated in Fig. 4.17. They are

visually similar

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Figure 4.12: MPP migration sequence of the input stacking pattern: a) the plain view; b) the perspective

view.

Figure 4.13: MPP migration sequence of the realization: a) the plain view; b) the perspective view.

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Figure 4.14: The sequential hierarchies of input and realization. The sequential hierarchy is quantified by

the dendrogram with the sequential cumulated distance. The hierarchical similarity is defined by the

similarity between two dendrograms. a) The input stacking pattern; b) the realization.

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Figure 4.15 A dendrogram is a weighted binary tree with three components: topology, label, and weights.

a) topology: the bifurcation structure between leaves of the tree; b) label: links each leaf to a branch; c)

weights: the height at which two branches merge, implying the distance between two clusters represented

by the two branches.

Figure 4.16: Cophenetic distance: distance at which two leaves merge. In the example, the cophenetic

distance from Leaf 1 to Leaf 3 is 2.5.

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Figure 4.17: ECDFs of the cophenetic distance from the input lobe sequence and from the simulated

realization in Fig. 4.14.

4.3.2 Implicit Hierarchical Control in the CRW-based Mechanism

The similarity between the ECDFs of the cophenetic distances from the input and

from the realization in Fig. 4.17 is not simply a coincidence. In fact, the ECDFs of the

cophenetic distance of realizations generated with sequential cumulated distances were

implicitly controlled by the ECDF of the stepwise migration distance. During the

simulation with the CRW-based mechanism, the ECDF of the stepwise migration was

reproduced in model realizations. Hence, the ECDF of the cophenetic distance from the

simulated dendrogram generated with the sequential cumulated distance were also

constrained. From the perspective of agglomerative hierarchical clustering, the

cophenetic distance, or the interbranch distance, within a dendrogram plot is determined

by the linkage function chosen to estimate the intercluster distance. Therefore, it is

worthwhile to study the relationship between different linkage functions and the

resulting ECDF of the cophenetic distance while the sequential cumulated distance is

applied in agglomerative hierarchical clustering. The linkage function, as briefly

introduced in Chapter 3, is the core of hierarchical clustering algorithms. It is always

easier to introduce an algorithm with a synthetic example of a four-point sequence (Fig.

4.18). Hierarchical clustering with a sequential cumulated distance (Eq. (3.6)) starts

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from grouping the closest pair of points in the given sequence. In Step 3, two groups,

{1,2} and {3,4}, are already formed since the sequential cumulated distance between

1,2 and 3,4 is the shortest. Before Step 4, the distance between the big groups, {1,2}

and {3,4}, should be calculated using the linkage function, which is consequently the

cophenetic distance between leaf 1 (or 2) and 3 (or 4).

A formal definition of the linkage function is a measure of dissimilarity between

two clusters (groups of observations) (Hastie, Tibshirani, and Friedman 2009). Various

definitions of linkage functions are proposed in the literature, from which the three most

prevalent types, 1) average linkage, 2) complete linkage, and 3) single linkage, are

briefly studied here. Let 𝐺 and 𝐻 be two groups between which the dissimilarity

𝑑(𝐺, 𝐻) will be measured. Let 𝑑𝑖𝑖′ represents the pairwise dissimilarity between

elements of 𝐺 and 𝐻, where one member of the pair 𝑖 is from Group 𝐺 and the other

member 𝑖′ is from Group 𝐻. The average linkage (Eq. (4.7)) defines the dissimilarity of

𝐺 and 𝐻 as the average of the pairwise dissimilarity of each member. The complete

linkage (Eq. (4.8)) calculates the intercluster dissimilarity to be that of the furthest pair

of elements. On the contrary, the single linkage (Eq. (4.9)) uses the dissimilarity of the

nearest pair of elements as the intercluster dissimilarity. 𝑑𝑖𝑖′ in Eq. (4.7) – Eq. (4.8) is

calculated with any non-sequential distance. Figure 4.19 shows dendrograms with

average, complete, and single linkages in the experimental and realization sequences in

Fig. 4.12 – 4.13. Comparing each column of Fig. 4.19, the observation is that the

application of different linkage functions significantly changes the structure of resulting

dendrograms. However, no significant hierarchical dissimilarity is observed between

dendrograms of the input and realization in each row. In general studies of the linkage

function in which the sequential cumulated distance is not used, imprecise relationships

(Eq. (4.10) – (4.11)) exist between the original distance (sequential cumulated distance

in this dissertation) and the cophenetic distances generated by different linkage

functions. A single linkage function generates dendrograms with the shortest

intercluster distance, which is even less or equal to the original distance. The complete

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linkage function produces the largest cophenetic distance that may be greater than the

original distance. The average linkage function ranges between the single and the

complete linkage functions. However, no clear relationship between cophenetic distance

generated with the average linkage function and the original distance is found in the

related algorithmic literature.

𝑑𝐴𝑣𝑒𝑟𝑎𝑔𝑒(𝐺, 𝐻) =1

𝑁𝐺𝑁𝐻∑ ∑ 𝑑𝑖𝑖′𝑖′∈𝐻𝑖∈𝐺 ( 4.7 )

𝑑𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒(𝐺, 𝐻) = 𝑚𝑎𝑥𝑖∈𝐺,𝑖′∈𝐻

𝑑𝑖𝑖′ ( 4.8 )

𝑑𝑆𝑖𝑛𝑔𝑙𝑒(𝐺, 𝐻) = 𝑚𝑖𝑛𝑖∈𝐺,𝑖′∈𝐻

𝑑𝑖𝑖′ ( 4.9 )

𝑑𝑆𝑖𝑛𝑔𝑙𝑒(𝐺, 𝐻) ≤ 𝑑𝐴𝑣𝑒𝑟𝑎𝑔𝑒(𝐺, 𝐻) ≤ 𝑑𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒(𝐺, 𝐻) ( 4.10 )

𝑑𝑆𝑖𝑛𝑔𝑙𝑒(𝐺, 𝐻) ≤ 𝑑𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙(𝐺, 𝐻) ≤ 𝑑𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒(𝐺, 𝐻) ( 4.11 )

While utilizing the sequential cumulated distance (Eq. (3.6)) in the hierarchical

clustering, equations of different linkage functions are simplified. Assuming the

temporal indices is 𝑚𝐺 ≤ 𝑖 ≤ 𝑛𝐺 within Group G, 𝑚𝐻 ≤ 𝑖 ≤ 𝑛𝐻 within Group 𝐻, and

𝑛𝐺 ≤ 𝑚𝐻, the relationship between the ECDFs of the cophenetic distances from the

results of single and complete linkage functions are defined by Eq. (4.12) – (4.13),

where 𝑑𝑚𝐺,𝑛𝐻 is computed with Eq. (3.6). Since the sequential cumulated distance

converges the problem into a one-dimensional geological space, the single linkage

function defines cophenetic distance from the last observation in the earlier group to the

first observation in the latter group, and the complete linkage function defines

cophenetic distance from the first observation in the earlier group to the last observation

in the latter group. The relationship in Eq. (4.10) – Eq. (4.11) is maintained in this

condition. The distance in complete linkage requires calculating the maximum of the

pairwise distance, which is the maximum of the sequential cumulated summations. Thus,

the complete linkage produces cophenetic distance measurements much greater than the

original distance. The average linkage produces a cophenetic distance that is between

the maximum and the minimum. As a demonstration, the L1-norm distance is applied

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to the ECDFs of the cophenetic distance to compare dendrograms of each row in Fig.

4.19, in which the distance is 3758.4 for average linkage, 15666 for complete linkage,

and 899.3 for single linkage, consistent with the analysis above.

Eq. (4.10) – Eq. (4.11) also define the imprecise relationship between the original

stepwise MPP migration distance and the cophenetic distance from the simulated

dendrogram. Thus, implicit control on the finalized lobe hierarchies of realizations

through the forward simulation process of the CRW-based MPP migration mechanism

is proved. In terms of reflecting the relative hierarchical similarity between two lobe

sequences, all linkage functions are the same as long as the same function is applied to

an ensemble to be compared. However, the single linkage is recommended since it

produces a cophenetic distance that is the closest to the original distance.

𝑑𝑆𝑖𝑛𝑔𝑙𝑒(𝐺, 𝐻) = 𝑑𝑛𝐺,𝑚𝐻 ( 4.12 )

𝑑𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒(𝐺, 𝐻) = 𝑑𝑚𝐺,𝑛𝐻 ( 4.13 )

Figure 4.18: The linkage function, a strategy of calculating the intercluster distance in the agglomerative

hierarchical clustering, determines the ECDF of the cophenetic distance. An imprecise relationship exists

between the ECDF of the stepwise migration distance and the ECDF of the cophenetic distance from the

dendrogram.

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Figure 4.19: The denrograms generated with different linkage functions using the input lobe sequence in

Fig. 4.12 and the realization in Fig. 4.13. The visual comparison can easily identify the difference between

the dendrograms with different linkage function on the same set of data. However, no significant

difference is observed between dendrograms of the input and realizations with the same linkage function.

The L1-norm differences between the ECDFs of the cophenetic distances of the input and realization

dendrograms are calculated for each linkage function: 3758.4 for complete linkage, 1566.6 for average

linkage, and 899.3 for single linkage, which is consistent with the theoretic analysis.

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4.3.3 Application of Hierarchical Similarity Control

The hierarchical similarity is a meaningful advantage for extending the

application of surface-based models. Updating of conceptual geological models with

additional data is always an important step at the appraisal stage of a hydrocarbon

reservoir. In a recently proposed strategy for rapidly updating conceptual models (Caers

2013), training images, which is the realization of a process-based model are used in

multiple-point statistics. Training images are quantified conceptual models, and the

updating is implemented by adding new training images or updating the probabilities of

each training image. It is also valuable to test the application of surface-based models

in this strategy for cases where training images are unavailable or solving a process-

based model is not allowed by the limited time of a project. Analogous to training

images in multiple-point statistics, an input lobe migration sequence is a quantification

of prior knowledge about the dynamic depositional process. We will further

demonstrate that different input sequences can effectively distinguish MPP hierarchies

in all model realizations, such that surface-based models can be directly used in the

rapid updating workflow as an alternative to multiple point statistics.

An algorithm comparison method based on an analysis of distance (Tan,

Tahmasebi, and Caers 2013) was applied to visualize differences between hierarchies

of realizations generated by models with conventional conceptual depositional rules and

with the CRW-based lobe migration mechanism. First, 210 realizations with different

lobe shapes were generated with the model built in Chapter 2, including 70 respective

realizations for narrow, medium, and fat lobes. Then, two different input lobe sequences

were chosen and 210 realizations of different lobe shapes were generated with each of

the sequences. One input sequence was the 38 lobe case from Exp. B, while the other

was a 296 lobe case from Exp. A. The lobe size in all realizations was constant, and the

model setting is demonstrated in Fig. 4.7. With the hierarchical similarity measure

defined in Section 4.3.1 – 4.3.2, MPP hierarchies of all realizations are plotted with the

first two components of multiple dimensional scaling (Scheidt and Caers 2009) in Fig.

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4.20. Dots in the plot represent all model realizations, and crossings represent two input

lobe sequences. Realizations with the conventional conceptual deposition mechanism

are very difficult to update because the prior knowledge is quantified by tens of

empirical coefficients, while realizations with the new CRW mechanism can be

effectively distinguished by simply changing the input lobe sequences. Although the

control of MPP hierarchy is imprecise, the method demonstrates a new methodology of

improving surface-based models.

Figure 4.20: First two component multidimensional scaling plot of hierarchies of realizations,

demonstrating the effect of an input lobe sequence on constraining the MPP hierarchies of model

realizations, which is difficult with the model built in Chapter 2.

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4.4 Chapter Summary

The high p-value stacking patterns were treated as the quantified prior information

to control the spatial distribution of lobes. Instead of characterizing information from

the final stacking patterns, a lobe stacking pattern is treated as a lobe migration

sequence. Assuming the lobe migration process is homogeneous and stationary at the

spatial and temporal scale of the lobe sequence, the migration of the lobe MPP is

characterized by the ECDF of the stepwise MPP migration distance and the MPP

migration orientation-shifting angle. With the ECDFs as inputs, the migration sequence

as CRW was described. The reason for choosing a technique as simple as CRW is that

it was desirable to control the number of parameters in the implementation, such that

the difficulty of conditioning a surface-based model to wells and thickness data was

reduced. Since only two dimensional horizontal movement data were available for the

stacking pattern, and both the migration movement of the input lobe sequence and the

conceptual rule of compensational stacking should be accounted for, a bifactor scheme

was designed such that the final movement of the MPP was determined by the weighted

average of the input lobe sequence and the compensational stacking rule. Once the

location of a MPP was calculated, the lobe orientation was obtained by a lobe shifting

angle from the overall basin flow orientation. The lobe shifting angle also followed its

statistics from the input lobe sequence. The first advantage of this implementation is the

simplicity. Essentially, aside from the ECDFs from inputs, only two empirical

coefficients were employed to produce realistic movement of MPPs, including

clustering, progradation, retrogradation, compensational stacking etc., which is

significantly reduced compared to the model built in Chapter 2. Second, CRW is an

implicit control of the hierarchies of realizations. To prove this, a quantitative

dissimilarity measure based on the cophenetic distance was proposed to quantify the

hierarchical similarity between the dendrograms of two lobe sequences. There was no

attempt to account for other types of spatial pattern information of the input lobe

sequence, such as polygonal proximity and shape proximity.

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The problem of modeling lobe migration is a spatial-temporal random process,

which should be ideally modeled as a function between the intermediate topography and

the spatial attributes of the next lobe, including location, orientation, polygonal pairwise

proximity and shape, in a three dimensional space. Essentially, the bifactor modeling

scheme is but a temporary solution because the lack of intermediate topographic

information in lobe sequences. However, the idea of a CRW-based implementation is

still a practical method to three-dimensional data.

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Chapter 5

CONCLUSIONS AND FUTURE WORK

5.1 Conclusions

Solutions to three problems were developed in this dissertation. First, the lack of

intermediate topographic information is a challenge in the surface-based modeling

approach. However, the data is difficult to obtain in real-scale analogues or active real

depositional systems. Geomorphic experiments were proposed to be a knowledge

database for static reservoir modeling (Chapter 2), since an experiment can produce

more complex depositional features than process-based models and can be recorded

comprehensively. The solution for this problem includes a method of extracting

intermediate geometric information from experimental records and a surface-based

model of distributary channel-lobe systems with the intermediate information as inputs.

Second, since the most similar experiment provides the most referable information, a

practical appraisal project aimed at specifying one experiment (among multiple options)

the most similar to a real depositional basin under appraisal. A statistical solution was

devised based on the spatial similarity between lobe stacking patterns from the

experiment and from the real system (Chapter 3). The solution consists of a set of

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algorithms to quantify geological concepts, such as lobe hierarchies, and algorithms to

analyze the concepts, which are also useful in comparative studies on lobate

environments. The last problem was how to honor the information of an experimental

sequence of lobes in the simulation of surface-based models. A new modeling strategy

was described in which all model realizations were hierarchically similar to the

identified lobe sequence. A quantitative measure of the hierarchical similarity between

lobe sequences was introduced (Chapter 4).

The integration of geomorphic experimental data in surface-based models in

Chapter 2 was initially proposed to improve erosion on shale drapes in deepwater lobate

systems. In terms of conceptual models in deepwater lobate environments, thin

subseismic scale shale drapes appear between sandy lobate bodies at different scales of

the stratigraphy and behave as flow barriers in a reservoir. Erosion on shale drapes

removes shale layers and causes directly contacted large sand bodies, thus impacting

the performance of a reservoir. Essentially, erosion caused by subsequent lobes requires

understanding the intermediate topography of the depositional process; however, the

intermediate topographic information is removed in real-scale outcrops and measuring

intermediate topography in active depositional systems is infeasible. Thus, erosion is

described with imprecise conceptual rules in current surface-based models. The

conceptual rules introduce large numbers of empirical coefficients, thus increasing

model complexity and workloads in uncertainty estimations. The geomorphic

experiment used in Chapter 2 includes intermediate topographic information measured

at fine temporal scales. Erosion and deposition are identified, extracted, and visualized

from experimental intermediate topographies, through which statistics of erosion-

deposition are interpreted. Since the visualized intermediate information implies

correlated erosion-deposition events, the lobate model was designed with negative-

positive surfaces. The negative-positive surfaces represent erosion and deposition,

respectively. The correlation between erosion and deposition geometries is

characterized by a dimensionless ratio between the maximum erosion depth and the

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maximum deposition thickness, such that information from small experiments is

scalable to large real systems.

To identify an experiment that is the most similar to a specific real depositional

system (Chapter 3), physical and conventional stratigraphic methods are not applicable

in experiments. The reason is that physical quantities of ancient sediment flows are not

measurable, and petrophysical properties used as the basis of interpretation in

conventional stratigraphy are simplified in geomorphic experiments. The proposed

solution attempts to find a statistical similarity between lobe stacking patterns from

different systems. A lobe stacking pattern is characterized by the ECDFs of the lobe

proximity parameters, and the similarity between two stacking patterns is estimated by

a p-value obtained from a two-sample hypothesis test on whether two empirical

distribution functions are from the same underlying population distribution. If two

stacking patterns are from the same underlying distribution, a reasonable assumption is

that the two processes are mutual references and information from the experiment is

applicable in modeling the real system. Lobes can be identified at different scales within

a lobate system. Thus, the solution provides a method to identify the proper scale of

interpretation in the experiment, at which the lobe pattern is more similar to a specific

real system than at other scales. Hierarchical clustering with constraints is applied on

parameterized lobe patterns so clustering is forced to follow the order of the lobe

sequence. Interpretation of large scale lobes from small scale lobes is automatically

performed with the algorithm. The significance of this solution is not limited to linking

experiments to real-scale systems, but also to quantitative definitions of lobe hierarchy

and methods applicable to the quantitative comparison of different experiments.

Lobe stacking patterns identified with the aforementioned method are quantified

scenarios of the depositional dynamic process. The concept of a geological dynamic

process is proposed as an extension of depositional rules to define forward processes of

distributing interpreted geobodies in response to the intermediate topography. Surface-

based models with lobe sequence inputs are the proper way to describe depositional

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processes. The integration of input lobe sequences in surface-based models includes

characterizing and simulating geologically meaningful statistics of the input sequence

with an algorithmically simple mechanism (Chapter 4). The input lobe sequence is

characterized by the stepwise relative migration distance and the migration orientation-

shifting angle based on the assumption that the geological dynamic process is stationary

in an input lobe sequence. The cumulative distribution functions of the lobe migration

distance and lobe migration orientation-shifting angle are used to control a CRW

describing locations of lobe MPPs. Since only two-dimensional horizontal patterns are

provided in the input lobe sequence while the compensational stacking rule has to be

accounted for as a conceptual rule in the simulation, a bifactor scheme is devised to

combine impacts from both the input lobe sequence and the conceptual compensational

stacking rules. With a quantitative definition of the lobe hierarchical similarity, it is

demonstrated that the CRW mechanism is an implicit control on the hierarchy of the

𝑀𝑃𝑃, although the control is imprecise.

5.2 Recommendations for Future Work

Distinguished from other static modeling techniques, the surface-based method is

the only means of explicitly modeling a forward geological dynamic process, instead of

non-forward statistical methods and forward physical methods. Explicitly considering

the forward geological dynamic process produces realistic realizations and empowers a

surface-based model to be the only method capable to study uncertainties caused by a

geological dynamic process, which is the essential determinant of spatial heterogeneity

of petrofacies. However, a large number of empirical coefficients in conceptual

depositional models limits applications of estimating uncertainty for geological

processes. With sufficient data provided in geomorphic experiments, design and

calibration of precise mathematical models for geological dynamic processes becomes

possible. This work is dedicated to the fundamental work of the quantification of

geological concepts, the characterization of geological dynamic processes, and the

125

simulation with constraints in geological processes. Suggestions for future work to

broaden and improve this work is discussed below.

Quantitative Assessment Criteria for Surface-based Algorithms

A most frequent question in assessing surface-based models is how to determine

that one implementation of a surface-based model is better than another. Generally, the

assessment of algorithms is based on the purpose of models. As indicated earlier, simple

and fast algorithms are better for uncertainty estimations while more complex, but

realistic, algorithms are more proper for testing hypotheses about depositional processes.

However, quantitative estimations of the comparison results are expected for serious

research. In this dissertation, geological concepts of interest were quantitatively defined

and then compared with the reproduction of these concepts with inputs, such as the

hierarchical similarity and lobe stacking pattern similarity. The comparison criterion of

algorithms can definitely be quantified with the degree of reproduction of inputs in the

realizations. For future work, alternative quantification of geological concepts may be

studied and compared to the dendrogram proposed in this dissertation. For example, an

alternative popular scale-dependent measure is Ripley’s K function. In spatial-temporal

processes, geological concepts of lobe migration are normally too complex to be

characterized by a single descriptor. A comparable study on advantages and

disadvantages of both techniques are valuable. The objective would be to understand

differences and relationships between different measures of geology and to implement

a surface-based lobate model constrained in all measures. Or, a new statistical measure

that combined advantages of both techniques could be devised.

The Relationship between Reservoir Performances and Dynamic

Geological Processes

Reservoir performances are affected by the heterogeneity of petrofacies, which is

further determined by the geological dynamic process sequentially distributing

interpreted regular shape geobodies of petrofacies. Hence, a reasonable hypothesis is

126

that the geological dynamic process is sensitive to the heterogeneity of petrofacies. Thus,

the direct uncertainty estimation of reservoir performances with parameters and

coefficients of a geological dynamic model may be more effective than using the

finalized models, such as training images. As indicated in Chapter 1, surface-based and

process-based models have been applied to select the most possible training images.

However, it is still valuable to study the relationship with the improved models, for

example, studying whether similarities will also be observed in the spatial distribution

of petrofacies given that hierarchies are statistically reproduced in realizations of the

CRW-based mechanism.

Identification of Three-Dimensional Geology with Two-Dimensional

Measurements

One assumption of Chapter 3 is that the horizontal spatial-temporal distributions

of real lobes are interpretable from field data. However, this assumption is strong

because such a spatial-temporal interpretation may not always be available. A weaker,

but still valuable, assumption will be that only several vertical sections of interpreted

channels and lobes are available from the field. If only several two-dimensional

geologies are available, identification of the three-dimensional geology, such as the

three-dimensional lobe hierarchy, will be significantly more complicated. More

uncertainties are involved in the problem, such as the number and the placement of cross

sections, etc. The hierarchically constrained CRW-based mechanism in this study may

be a starting point to study this problem. Given that an ensemble of realizations with

similar lobe hierarchies are generated, hierarchies of two-dimensional sections of lobes

may be available from various numbers of vertical cross sections obtained at different

locations. Thus, the statistical relationship between hierarchies within two-dimensional

realizations and the input patterns may be estimated. The challenges will still be huge

uncertainties associated with two-dimensional sections.

127

Simulation of Subseismic Stratigraphies with Known Hierarchies in

Experiments

One of the original interests of static reservoir modeling is the reconstruction of

stratigraphies at subseismic scales. Since the experiment equivalent scale of lobes in the

real system was identified with the workflow in Chapter 3 and the experimental lobe

hierarchy is available, it is reasonable to utilize hierarchical information in the

experiment to model subseismic scale lobate bodies in the real environment. The

primary challenge of this problem is how to control the overall hierarchy of lobes in the

forward simulation scheme of surface-based models with an efficient implementation.

Thorough Utilization of the Experimental Data

The dissertation presented a simple implementation of a lobe migration

mechanism to demonstrate improvements led by the integration of experimental data in

the surface-based approach. However, more possible improvements can be expected if

the experimental data are more thoroughly utilized.

i) Three-dimensional intermediate information

The prerequisite of modeling geological dynamic processes is the availability of

three-dimensional intermediate topographies measured at a sufficient temporal

resolution. Temporary solutions are implemented in this dissertation since the three-

dimensional intermediate topography is not available in the experiments. Once the

three-dimensional information is available, possible improvements are expected as

follows: First, three-dimensional geometries of erosion and deposition can be extracted,

so the correlation between the real erosion and deposition surfaces can be modeled more

precisely than a general thickness function and a dimensionless ratio. Second, the noisy

scale versus p-value plots for the polygonal similarity and shape similarity in Chapter 3

may exist either because of the improper measures or because of the insufficiency of

describing the lobe with simply two-dimensional patterns. Thus, more complex

128

polygonal distance or shape dissimilarity measures are worth studying with three-

dimensional geometries. Finally, the bifactor modeling scheme developed in Chapter 4

is no longer needed with three-dimensional intermediate information, since the

conceptual compensational stacking rule can be precisely quantified. The CRW-based

characterization and implementation can be extended to the three-dimensional space

and is thus a good starting point. Lobe orientation, which is determined by the regional

intermediate topography, can be quantitatively characterized with respect to

intermediate topography instead of a relative angle apart from an interpreted basin flow

orientation.

ii) Hierarchical simulation on more parameters

The presented CRW-based implementation of the lobe migration mechanism only

constrains the hierarchy of MPPs in model realizations. As indicated earlier, it is optimal

to maximize the use of input sequences by constraining hierarchies for all the parameters.

However, the challenge is that implementations are expected to control the global

hierarchy in a simple and efficient forward simulation scheme with the least number of

empirical coefficients.

iii) Characterization and simulation of distributary channels

Distributary channels are also observed in experiments, thus channel sequences

are sufficient for developing precise quantitative measures for channel geometry and

channel hierarchy. Compared to lobes, channels are more complex in both morphology

and hierarchy. Channel morphology is a stochastic function of intermediate topography.

The stochastic function varies along the channel centerline toward the downstream

direction. Hierarchy is determined by avulsion points, thus the number, location, and

migration of avulsion points may be studied for modeling channel hierarchies.

Techniques are expected to characterize, compare, and simulate this higher order

problem.

129

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